Samacheer Kalvi Class 10 Maths Solutions Chapter 7 Mensuration More Ques

Tamilnadu Samacheer Kalvi 10th Maths Solutions Chapter 7 Mensuration Additional Questions

I. Multiple Choice Questions:

 

Question 1. The curved surface area of a right circular cylinder of radius 1 cm and height 1 cm is equal to ______
(a) π cm²
(b) 2π cm²
(c) 3π cm²
(d) 2 cm²
Answer: (b) 2π cm²
In simple words: The curved surface area of a cylinder is found by multiplying 2, pi, the radius, and the height. When the radius and height are both 1 cm, the area is 2π square centimeters.

🎯 Exam Tip: Remember the formula for the curved surface area of a cylinder, \( 2\pi rh \), and be careful with units. Always double-check your calculations.

 

Question 2. The total surface area of a solid right circular cylinder whose radius is half of its height h is equal to ______ sq. units.
(a) \( \frac{3}{2} \pi h \)
(b) \( \frac{2}{3} \pi h^{2} \)
(c) \( \frac{3}{2} \pi h^{2} \)
(d) \( \frac{2}{3} \pi h \)
Answer: (c) \( \frac{3}{2} \pi h^{2} \)
In simple words: If a cylinder's radius is half its height, its total outer surface area can be written using only the height, h, in the formula \( \frac{3}{2} \pi h^{2} \). We calculate this by using \( r = h/2 \) in the total surface area formula \( 2\pi r(h+r) \).

🎯 Exam Tip: When the dimensions are given in terms of each other, substitute the relationship (like \( r = h/2 \)) into the formula early to simplify the expression. This makes it easier to compare with the options.

 

Question 3. Base area of a right circular cylinder is 80 cm². If its height is 5 cm, then the volume is equal to ______
(a) 400 cm³
(b) 16 cm³
(c) 200 cm³
(d) 300 cm³
Answer: (a) 400 cm³
In simple words: To find the volume of a cylinder, you multiply its base area by its height. Since the base area is 80 cm² and height is 5 cm, the volume is 400 cubic centimeters.

🎯 Exam Tip: Remember that the base area of a cylinder is \( \pi r^2 \), so the volume formula \( \pi r^2 h \) can also be thought of as "base area × height". This simplifies calculations when the base area is directly given.

 

Question 4. If the total surface area of a solid right circular cylinder is 200π cm² and its radius is 5 cm, then the sum of its height and radius is ______
(a) 20 cm
(b) 25 cm
(c) 30 cm
(d) 15 cm
Answer: (a) 20 cm
In simple words: We use the formula for total surface area, \( 2\pi r(h+r) \). By putting in the given total surface area and radius, we can directly find the sum of the height and radius.

🎯 Exam Tip: The formula for total surface area, \( 2\pi r(h+r) \), directly includes the term \( (h+r) \), which simplifies finding their sum. Isolate this term to quickly solve the problem.

 

Question 5. The curved surface area of a right circular cylinder whose radius is a units and height is b units, is equal to ______
(a) πa²b sq.cm
(b) 2πab sq.cm
(c) 2π sq.cm
(d) 2 sq.cm
Answer: (b) 2πab sq.cm
In simple words: The formula for the curved surface area of a cylinder is \( 2\pi \times \text{radius} \times \text{height} \). If the radius is 'a' and height is 'b', then the area is simply \( 2\pi ab \). This shows how algebraic terms represent real-world dimensions in formulas.

🎯 Exam Tip: Clearly identify the given radius and height from the problem statement and substitute them into the correct formula for the curved surface area of a cylinder, which is \( 2\pi rh \).

 

Question 6. Radius and height of a right circular cone and that of a right circular cylinder are respectively, equal. If the volume of the cylinder is 120 cm³, then the volume of the cone is equal to ______
(a) 1200 cm³
(b) 360 cm³
(c) 40 cm³
(d) 90 cm³
Answer: (c) 40 cm³
In simple words: The volume of a cone is exactly one-third the volume of a cylinder if they have the same base radius and height. So, if the cylinder's volume is 120 cm³, the cone's volume will be 40 cm³.

🎯 Exam Tip: Remember the fundamental relationship: \( V_{cone} = \frac{1}{3} V_{cylinder} \) when their radii and heights are identical. This is a common shortcut in Mensuration problems.

 

Question 7. If the diameter and height of a right circular cone are 12 cm and 8 cm respectively, then the slant height is ______
(a) 10 cm
(b) 20 cm
(c) 30 cm
(d) 96 cm
Answer: (a) 10 cm
In simple words: To find the slant height of a cone, we use the Pythagorean theorem because the height, radius, and slant height form a right-angled triangle. Since the diameter is 12 cm, the radius is 6 cm. Using \( l = \sqrt{r^2 + h^2} \), we get 10 cm.

🎯 Exam Tip: Always remember that diameter is twice the radius. Apply the Pythagorean theorem \( l^2 = r^2 + h^2 \) (or \( l = \sqrt{r^2 + h^2} \)) for slant height calculations.

 

Question 8. If the circumference at the base of a right circular cone and the slant height are 120π cm and 10 cm respectively, then the curved surface area of the cone is equal to ______
(a) 1200π cm²
(b) 600π cm²
(c) 300π cm²
(d) 600 cm²
Answer: (b) 600π cm²
In simple words: First, find the radius using the circumference formula \( C = 2\pi r \). Then, use the curved surface area formula for a cone, \( CSA = \pi r l \), to find the answer.

🎯 Exam Tip: The curved surface area of a cone can be calculated as \( \frac{1}{2} \times \text{base circumference} \times \text{slant height} \). This shortcut often saves a step in such problems.

 

Question 9. If the volume and the base area of a right circular cone are 48π cm³ and 12π cm² respectively, then the height of the cone is equal to ______
(a) 6 cm
(b) 8 cm
(c) 10 cm
(d) 12 cm
Answer: (d) 12 cm
In simple words: The volume of a cone is given by \( \frac{1}{3} \times \text{base area} \times \text{height} \). If we know the volume and the base area, we can easily find the height by rearranging the formula.

🎯 Exam Tip: Understand that \( \pi r^2 \) represents the base area. The formula for the volume of a cone \( V = \frac{1}{3} \pi r^2 h \) can be rewritten as \( V = \frac{1}{3} \times (\text{Base Area}) \times h \), making it direct to find height.

 

Question 10. If the height and the base area of a right circular cone are 5 cm and 48 sq.cm respectively, then the volume of the cone is equal to ______
(a) 240 cm³
(b) 120 cm³
(c) 80 cm³
(d) 480 cm³
Answer: (c) 80 cm³
In simple words: We can find the volume of the cone by multiplying one-third of the base area by its height. Since the base area is 48 sq.cm and height is 5 cm, the volume is 80 cubic centimeters.

🎯 Exam Tip: Remember the volume formula for a cone \( V = \frac{1}{3} \times \text{Base Area} \times \text{height} \). This makes it very easy when the base area is directly provided.

 

Question 11. The ratios of the respective heights and the respective radii of two cylinders are 1 : 2 and 2 : 1 respectively. Then their respective volumes are in the ratio ______
(a) 4:1
(b) 2:1
(c) 1:2
Answer: (b) 2:1
In simple words: The volume of a cylinder depends on the square of its radius and its height. If we use the given ratios for radius and height for two cylinders, we find their volumes are in a 2:1 ratio.

🎯 Exam Tip: The ratio of volumes of two cylinders \( V_1 : V_2 \) is \( r_1^2 h_1 : r_2^2 h_2 \). Substitute the given ratios for radii and heights directly into this relationship to quickly find the volume ratio.

 

Question 12. If the radius of a sphere is 2 cm, then the curved surface area of the sphere is equal to ______
(a) 8π cm²
(b) 16 cm²
(c) 12π cm²
(d) 16π cm²
Answer: (d) 16π cm²
In simple words: The curved surface area of a sphere is \( 4\pi r^2 \). If the radius is 2 cm, then the area is \( 4\pi \times (2)^2 = 16\pi \) square centimeters. A sphere has only one continuous surface, so its curved surface area is the same as its total surface area.

🎯 Exam Tip: For a sphere, the curved surface area is the same as its total surface area, which is \( 4\pi r^2 \). Make sure to square the radius correctly.

 

Question 13. The total surface area of a solid hemisphere of diameter 2 cm is equal to ______
(a) 12π cm²
(b) 4π cm²
(c) 3π cm²
Answer: (c) 3π cm²
In simple words: For a solid hemisphere, the total surface area includes both the curved part and its flat circular base. If the diameter is 2 cm, the radius is 1 cm. The total surface area is \( 3\pi r^2 \), which calculates to \( 3\pi \) square centimeters.

🎯 Exam Tip: Remember the total surface area formula for a solid hemisphere is \( 3\pi r^2 \) (which is \( 2\pi r^2 \) for the curved part plus \( \pi r^2 \) for the base). Do not confuse it with the curved surface area of a hemisphere (\( 2\pi r^2 \)) or a full sphere (\( 4\pi r^2 \)).

 

Question 14. If the volume of a sphere is \( \frac{9}{16} \pi \) cu.cm, then its radius is ______
(a) \( \frac{4}{3} \) cm
(b) \( \frac{3}{4} \) cm
(c) \( \frac{3}{2} \) cm
(d) \( \frac{2}{3} \) cm
Answer: (b) \( \frac{3}{4} \) cm
In simple words: We use the volume formula for a sphere, \( V = \frac{4}{3} \pi r^3 \), and set it equal to the given volume. Then, we solve for 'r' by isolating \( r^3 \) and taking the cube root.

🎯 Exam Tip: Be careful with algebraic manipulation when solving for \( r^3 \). Remember to take the cube root of both the numerator and the denominator separately if it's a fraction.

 

Question 15. The surface areas of two spheres are in the ratio of 9 : 25. Then their volumes are in the ratio ______
(a) 81: 625
(b) 729: 15625
(c) 27: 125
Answer: (c) 27: 125
In simple words: If surface areas are in a ratio, their radii will be in the square root of that ratio. Since volume depends on the cube of the radius, the ratio of volumes will be the cube of the radii ratio.

🎯 Exam Tip: For similar 3D figures, if the ratio of their surface areas is \( a^2:b^2 \), then the ratio of their radii is \( a:b \), and the ratio of their volumes is \( a^3:b^3 \). This relationship is very useful for quick problem-solving.

 

Question 16. The total surface area of a solid hemisphere whose radius is a units, is equal to ______
(a) 2πα² sq. units
(b) 3πα² sq. units
(c) 3πa sq. units
(d) 3a² sq. units
Answer: (b) 3πα² sq. units
In simple words: The total surface area of a solid hemisphere includes its curved part (\( 2\pi r^2 \)) and its flat circular base (\( \pi r^2 \)). When the radius is 'a', the total area becomes \( 3\pi a^2 \) square units.

🎯 Exam Tip: Distinguish between the curved surface area (\( 2\pi r^2 \)) and the total surface area (\( 3\pi r^2 \)) of a solid hemisphere. The total includes the base area. For a hollow hemisphere, it would only be \( 2\pi r^2 \) for a thin shell or \( 2\pi(R^2+r^2) \) if considering inner and outer surfaces and thickness.

 

Question 17. If the surface area of a sphere is 100π cm², then its radius is equal to ______
(a) 25 cm
(b) 100 cm
(c) 5 cm
(d) 10 cm
Answer: (c) 5 cm
In simple words: Use the formula for the surface area of a sphere, \( SA = 4\pi r^2 \). Set the given surface area equal to this formula and solve for the radius, r.

🎯 Exam Tip: When working backwards from surface area to radius, remember to divide by \( 4\pi \) first, then take the square root to find 'r'.

 

Question 18. If the surface area of a sphere is 36π cm², then the volume of the sphere is equal to ______
(a) 12π cm³
(b) 36π cm³
(c) 72π cm³
(d) 108π cm³
Answer: (b) 36π cm³
In simple words: First, find the radius from the given surface area using \( SA = 4\pi r^2 \). Once the radius is found, use it in the volume formula for a sphere, \( V = \frac{4}{3} \pi r^3 \).

🎯 Exam Tip: This is a two-step problem. Always find the radius first using the surface area information, then use that radius to calculate the volume. Make sure not to mix up the formulas for surface area and volume.

 

Question 19. If the total surface area of a solid hemisphere is 12π cm² then its curved surface area is equal to ______
(a) 6π cm²
(b) 24π cm²
(c) 36π cm²
(d) 8π cm²
Answer: (d) 8π cm²
In simple words: First, use the total surface area of a solid hemisphere \( (3\pi r^2) \) to find the radius. Once you have the radius, calculate the curved surface area of the hemisphere using the formula \( 2\pi r^2 \).

🎯 Exam Tip: This question requires you to distinguish between total surface area and curved surface area of a hemisphere. The total includes the base, the curved does not. Solve for radius using the total surface area, then apply it to the curved surface area formula.

 

Question 20. If the radius of a sphere is half of the radius of another sphere, then their respective volumes are in the ratio ______
(a) 1:8
(b) 2:1
(c) 1:2
(d) 8:1
Answer: (a) 1:8
In simple words: If one sphere's radius is half of another's, then the ratio of their radii is 1:2. Since the volume of a sphere depends on the cube of its radius, the ratio of their volumes will be \( 1^3 : 2^3 \), which is 1:8.

🎯 Exam Tip: For any two similar 3D solids, if the ratio of their corresponding linear dimensions (like radii) is \( a:b \), then the ratio of their volumes is \( a^3:b^3 \). Apply this property directly to solve such ratio problems quickly.

III. Answer the following questions.

 

Question 1. Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq.cm and 110 cm respectively. Find its height and diameter.
Answer: Given, the circumference of the base of a cylinder is 110 cm. This means \( 2\pi r = 110 \) cm. Also, the curved surface area is 4400 cm². This means \( 2\pi r h = 4400 \) cm². We know that the circumference of the base is part of the curved surface area formula.
From \( 2\pi r = 110 \) (Equation 1)
From \( 2\pi r h = 4400 \) (Equation 2)
Divide Equation 2 by Equation 1:
\( \frac{2\pi r h}{2\pi r} = \frac{4400}{110} \)
\( \implies h = 40 \) cm
So, the height of the cylinder is 40 cm.
Now, use Equation 1 to find the radius:
\( 2 \times \frac{22}{7} \times r = 110 \)
\( r = \frac{110 \times 7}{2 \times 22} \)
\( r = \frac{5 \times 7}{2} = \frac{35}{2} = 17.5 \) cm
The diameter \( (d) \) is twice the radius:
\( d = 2 \times r = 2 \times 17.5 = 35 \) cm
Therefore, the height of the cylinder is 40 cm and its diameter is 35 cm. The circumference provides the base radius, which is essential for relating to the curved surface area and then finding the height.
In simple words: First, we used the base circumference to find the radius. Then, we divided the curved surface area by the circumference to get the height. Finally, we doubled the radius to find the diameter.

🎯 Exam Tip: Always list all given values clearly. Identify which formulas relate these values and plan your steps: usually, one formula helps find an unknown, which is then used in another formula.

 

Question 2. A mansion has 12 right cylindrical pillars each having radius 50 cm and height 3.5 m. Find the cost of painting the lateral surface of the pillars at Rs 20 per square metre.
Answer: Given,
Radius of a cylinder \( (r) = 50 \) cm \( = 0.5 \) m (convert cm to m for consistency with cost per square metre)
Height of a cylinder \( (h) = 3.5 \) m
The curved surface area of one pillar (lateral surface) is \( 2\pi r h \) sq. units.
Curved surface area of one pillar \( = 2 \times \frac{22}{7} \times 0.5 \times 3.5 \text{ m}^2 \)
\( = 2 \times \frac{22}{7} \times \frac{1}{2} \times \frac{7}{2} \text{ m}^2 \)
\( = 22 \times \frac{1}{2} \text{ m}^2 = 11 \text{ m}^2 \)
Now, calculate the curved surface area for all 12 pillars:
Curved surface area of 12 pillars \( = 12 \times 11 \text{ m}^2 = 132 \text{ m}^2 \)
The cost of painting per square metre is Rs 20.
Total cost of painting \( = \text{Total area} \times \text{Cost per square metre} \)
Total cost of painting \( = 132 \times \text{Rs } 20 = \text{Rs } 2640 \)
So, the total cost of painting the lateral surface of the pillars is Rs 2640. Painting lateral surfaces is common in construction and helps protect materials from weather.
In simple words: First, we found the curved surface area of one pillar using the given radius and height. Then, we multiplied this by 12 for all pillars. Finally, we multiplied the total area by the painting cost per square meter to get the total cost.

🎯 Exam Tip: Ensure all units are consistent before calculating (e.g., convert cm to m). Remember to calculate the area for all objects (here, 12 pillars) before multiplying by the cost.

 

Question 3. The total surface area of a solid right circular cylinder is 231 cm². Its curved surface area is two thirds of the total surface area. Find the radius and height of the cylinder.
Answer: Given,
Total surface area \( (\text{T.S.A.}) \) of the cylinder \( = 231 \text{ cm}^2 \)
Curved surface area \( (\text{C.S.A.}) = \frac{2}{3} \times \text{T.S.A.} \)
\( \text{C.S.A.} = \frac{2}{3} \times 231 = 2 \times 77 = 154 \text{ cm}^2 \)
We know that \( \text{T.S.A.} = \text{C.S.A.} + 2 \times (\text{Base Area}) \)
\( \text{T.S.A.} = \text{C.S.A.} + 2\pi r^2 \)
Substitute the given values:
\( 231 = 154 + 2\pi r^2 \)
\( 2\pi r^2 = 231 - 154 = 77 \)
Now, solve for the radius \( (r) \):
\( 2 \times \frac{22}{7} \times r^2 = 77 \)
\( r^2 = \frac{77 \times 7}{2 \times 22} = \frac{7 \times 7}{2 \times 2} = \frac{49}{4} \)
\( r = \sqrt{\frac{49}{4}} = \frac{7}{2} = 3.5 \) cm
Now, use the curved surface area formula to find the height \( (h) \):
\( \text{C.S.A.} = 2\pi r h = 154 \)
\( 2 \times \frac{22}{7} \times 3.5 \times h = 154 \)
\( 2 \times \frac{22}{7} \times \frac{7}{2} \times h = 154 \)
\( 22 h = 154 \)
\( h = \frac{154}{22} = 7 \) cm
So, the radius of the cylinder is 3.5 cm and its height is 7 cm. The relationship between total surface area and curved surface area helps us isolate the area of the base circles.
In simple words: First, we calculated the curved surface area from the total surface area. Then, we used these to find the area of the two circular bases. From the base area, we found the radius. Lastly, we used the curved surface area and radius to find the height.

🎯 Exam Tip: Always write down the known formulas for total surface area (\( 2\pi r(h+r) \)) and curved surface area (\( 2\pi rh \)). The difference between these two will give you \( 2\pi r^2 \), which is key to finding the radius.

 

Question 4. The total surface area of a solid right circular cylinder is 1540 cm². If the height is four times the radius of the base, then find the height of the cylinder.
Answer: Given,
Total surface area \( (\text{T.S.A.}) \) of the cylinder \( = 1540 \text{ cm}^2 \)
Let the radius of the cylinder be 'r'.
According to the given condition, height \( (h) = 4r \).
The formula for total surface area of a cylinder is \( \text{T.S.A.} = 2\pi r (h+r) \).
Substitute the value of \( h = 4r \) into the formula:
\( 1540 = 2\pi r (4r+r) \)
\( 1540 = 2\pi r (5r) \)
\( 1540 = 10\pi r^2 \)
Now, solve for \( r \):
\( 1540 = 10 \times \frac{22}{7} \times r^2 \)
\( r^2 = \frac{1540 \times 7}{10 \times 22} \)
\( r^2 = \frac{154 \times 7}{22} \)
\( r^2 = 7 \times 7 = 49 \)
\( r = \sqrt{49} = 7 \) cm
Now, find the height using the relation \( h = 4r \):
\( h = 4 \times 7 = 28 \) cm
Therefore, the height of the cylinder is 28 cm. Understanding how dimensions relate (like height being a multiple of radius) simplifies the formulas needed.
In simple words: We used the relationship that height is four times the radius to write the total surface area formula in terms of radius only. Then, we solved for the radius and multiplied it by four to get the height.

🎯 Exam Tip: When given a relationship between dimensions (like \( h = 4r \)), substitute it into the formula at the beginning. This transforms the problem into solving for a single variable, making it much simpler.

 

Question 5. If the vertical angle and the radius of a right circular cone are 60° and 15 cm respectively, then find its height and slant height.
Answer: Let the cone be OAB, with O as the vertex and C as the center of the base. AC is the radius.
Given, the vertical angle of the cone is \( 60^\circ \).
In the right triangle \( \triangle OAC \), the semi-vertical angle \( \angle AOC = \frac{60^\circ}{2} = 30^\circ \).
Given the radius \( AC = 15 \) cm.
To find the height \( (OC) \):
In \( \triangle OAC \), we have \( \tan(\angle AOC) = \frac{AC}{OC} \)
\( \tan 30^\circ = \frac{15}{OC} \)
\( \frac{1}{\sqrt{3}} = \frac{15}{OC} \)
\( OC = 15\sqrt{3} \) cm
So, the height of the cone \( (h) = 15\sqrt{3} \) cm.
To find the slant height \( (OA) \):
In \( \triangle OAC \), we have \( \sin(\angle AOC) = \frac{AC}{OA} \)
\( \sin 30^\circ = \frac{15}{OA} \)
\( \frac{1}{2} = \frac{15}{OA} \)
\( OA = 15 \times 2 = 30 \) cm
So, the slant height of the cone \( (l) = 30 \) cm. Trigonometry is very useful for finding unknown dimensions in cones and other geometric shapes when angles are known.
In simple words: We used the semi-vertical angle (half of the vertical angle) and the radius to form a right-angled triangle. Then, using trigonometry (tan for height, sin for slant height), we calculated the height and slant height of the cone.

🎯 Exam Tip: When given the vertical angle of a cone, always remember to use the semi-vertical angle in your trigonometric calculations, as it forms a right-angled triangle with the height and radius. Ensure you know the basic trigonometric ratios.

 

Question 6. Radius and height of a right circular cone and that of a right circular cylinder are respectively, equal. If the volume of the cylinder is 120 cm³, then the volume of the cone is equal to
(1) 1200 cm³
(2) 360 cm³
(3) 40 cm³
(4) 90 cm³
Answer: (3) 40 cm³
In simple words: The volume of a cone is always one-third of the volume of a cylinder if both have the same base radius and height. So, if the cylinder's volume is 120 cm³, the cone's volume will be 40 cm³.

🎯 Exam Tip: Remember the key relationship that the volume of a cone is \( \frac{1}{3} \) the volume of a cylinder with the same base and height. This is a common formula application.

 

Question 7. If the diameter and height of a right circular cone are 12 cm and 8 cm respectively, then the slant height is
(1) 10 cm
(2) 20 cm
(3) 30 cm
(4) 96 cm
Answer: (1) 10 cm
In simple words: We find the slant height of the cone using a special triangle rule, like the Pythagoras theorem. With a diameter of 12 cm, the radius is 6 cm. Using the height of 8 cm, the slant height comes out to be 10 cm.

🎯 Exam Tip: Always remember that the radius, height, and slant height of a cone form a right-angled triangle, allowing you to use the Pythagorean theorem \( l^2 = r^2 + h^2 \) to find any missing side.

 

Question 8. If the circumference at the base of a right circular cone and the slant height are 120π cm and 10 cm respectively, then the curved surface area of the cone is equal to
(1) 1200π cm²
(2) 600π cm²
(3) 300π cm²
(4) 600 cm²
Answer: (2) 600π cm²
In simple words: We use the formula for the curved surface area of a cone, which is half of the base circumference multiplied by the slant height. Since the circumference is 120π cm and the slant height is 10 cm, the curved surface area becomes 600π cm².

🎯 Exam Tip: The curved surface area of a cone can be directly calculated as \( \frac{1}{2} \times \text{Circumference} \times \text{slant height} \). This shortcut helps save a step compared to finding the radius first.

 

Question 9. If the volume and the base area of a right circular cone are 48π cm³ and 12π cm² respectively, then the height of the cone is equal to
(1) 6 cm
(2) 8 cm
(3) 10 cm
(4) 12 cm
Answer: (4) 12 cm
In simple words: The volume of a cone is one-third of its base area multiplied by its height. We can use the given volume and base area to find the height. By dividing the volume by one-third of the base area, we get the height as 12 cm.

🎯 Exam Tip: The formula for the volume of a cone is \( V = \frac{1}{3} \times \text{Base Area} \times h \). This allows you to find height directly if volume and base area are known, without needing the radius.

 

Question 10. If the height and the base area of a right circular cone are 5 cm and 48 sq.cm respectively, then the volume of the cone is equal to
(1) 240 cm³
(2) 120 cm³
(3) 80 cm³
(4) 480 cm³
Answer: (3) 80 cm³
In simple words: To find the volume of a cone, multiply the base area by the height and then divide by three. So, with a base area of 48 sq. cm and a height of 5 cm, the volume is 80 cm³.

🎯 Exam Tip: For cones, always remember the volume formula \( V = \frac{1}{3} \pi r^2 h \). If the base area (\( \pi r^2 \)) is directly given, substitute it without calculating the radius separately.

 

Question 11. The ratios of the respective heights and the respective radii of two cylinders are 1 : 2 and 2 : 1 respectively. Then their respective volumes are in the ratio
(1) 4:1
(3) 2:1
(4) 1:2
Answer: (3) 2:1
In simple words: The volume of a cylinder depends on the square of its radius and its height. When we compare two cylinders, we can find the ratio of their volumes by multiplying the square of their radii ratios with their height ratio. For the given ratios, the volumes are in a 2:1 ratio.

🎯 Exam Tip: When dealing with ratios of volumes, remember that volume is proportional to \( r^2 h \). So, the ratio of volumes will be \( (r_1/r_2)^2 \times (h_1/h_2) \).

 

Question 12. If the radius of a sphere is 2 cm, then the curved surface area of the sphere is equal to
(1) 8π cm²
(2) 16 cm²
(3) 12π cm²
(4) 16π cm²
Answer: (4) 16π cm²
In simple words: The curved surface area of a sphere is found by a specific formula: four times pi multiplied by the radius squared. For a radius of 2 cm, the surface area comes out to be 16π cm².

🎯 Exam Tip: The formula for the surface area of a sphere is \( 4 \pi r^2 \). This is a fundamental formula to memorize for sphere problems.

 

Question 13. The total surface area of a solid hemisphere of diameter 2 cm is equal to
(2) 12π cm²
(3) 4π cm²
(4) 3π cm²
Answer: (4) 3π cm²
In simple words: A solid hemisphere has both a curved surface and a flat circular base. Its total surface area is three times pi multiplied by the radius squared. With a diameter of 2 cm, the radius is 1 cm, so the total surface area is 3π cm².

🎯 Exam Tip: For a solid hemisphere, the total surface area is \( 3 \pi r^2 \) (curved surface \( 2 \pi r^2 \) + base area \( \pi r^2 \)). Be careful not to confuse it with the curved surface area alone.

 

Question 14. If the volume of a sphere is \( \frac{9}{16} \pi \) cu.cm, then its radius is
(1) \( \frac{4}{3} \) cm
(2) \( \frac{3}{4} \) cm
(3) \( \frac{3}{2} \) cm
(4) \( \frac{2}{3} \) cm
Answer: (2) \( \frac{3}{4} \) cm
In simple words: We know the formula for the volume of a sphere. By setting the given volume equal to the formula, we can solve for the radius. We find that the radius is \( \frac{3}{4} \) cm.

🎯 Exam Tip: Always remember the volume formula for a sphere \( V = \frac{4}{3} \pi r^3 \). When working backwards from a given volume, isolate \( r^3 \) and then take the cube root.

 

Question 15. The surface areas of two spheres are in the ratio of 9 : 25. Then their volumes are in the ratio
(1) 81: 625
(2) 729: 15625
(4) 27: 125
Answer: (4) 27: 125
In simple words: The ratio of surface areas of two spheres is equal to the square of their radii ratio. So, if the surface area ratio is 9:25, the radii ratio is 3:5. The ratio of volumes of two spheres is equal to the cube of their radii ratio, which means the volume ratio is 27:125.

🎯 Exam Tip: For similar 3D shapes like spheres, if the ratio of linear dimensions (like radius) is \( a:b \), then the ratio of their surface areas is \( a^2:b^2 \) and the ratio of their volumes is \( a^3:b^3 \).

 

Question 16. The total surface area of a solid hemisphere whose radius is a units, is equal to
(1) 2πα² sq. units
(2) 3πα² sq. units
(3) 3πa sq. units
(4) 3a² sq. units
Answer: (2) 3πα² sq. units
In simple words: For a solid hemisphere, the total surface area includes its curved top and flat circular base. This adds up to three times pi multiplied by the square of its radius. If the radius is 'a', the area is 3πα² sq. units.

🎯 Exam Tip: Distinguish between the curved surface area \( (2 \pi r^2) \) and the total surface area \( (3 \pi r^2) \) for a solid hemisphere. The 'solid' implies the base is included.

 

Question 17. If the surface area of a sphere is 100π cm², then its radius is equal to
(1) 25 cm
(2) 100 cm
(3) 5 cm
(4) 10 cm
Answer: (3) 5 cm
In simple words: We use the formula for the surface area of a sphere, which is 4πr². By setting this equal to the given surface area of 100π cm², we can solve to find that the radius of the sphere is 5 cm.

🎯 Exam Tip: When surface area is given, use the formula \( 4 \pi r^2 \) to solve for the radius. Remember to simplify by dividing out \( \pi \) and then taking the square root.

 

Question 18. If the surface area of a sphere is 36π cm², then the volume of the sphere is equal to
(1) 12π cm³
(2) 36π cm³
(3) 72π cm³
(4) 108π cm³
Answer: (2) 36π cm³
In simple words: First, we use the given surface area to find the radius of the sphere. Once we have the radius, we can use the formula for the volume of a sphere to calculate it. The radius turns out to be 3 cm, and with that, the volume is 36π cm³.

🎯 Exam Tip: This question requires a two-step approach: first, calculate the radius from the given surface area, and then use that radius to calculate the volume. Make sure to use the correct formulas for each step.

 

Question 19. If the total surface area of a solid hemisphere is 12π cm² then its curved surface area is equal to
(1) 6π cm²
(2) 24π cm²
(3) 36π cm²
(4) 8π cm²
Answer: (4) 8π cm²
In simple words: The total surface area of a solid hemisphere is \( 3 \pi r^2 \), and its curved surface area is \( 2 \pi r^2 \). Given the total surface area, we can find the radius and then calculate the curved surface area. We find that the radius is 2 cm, making the curved surface area 8π cm².

🎯 Exam Tip: Remember the relationship: Total Surface Area (TSA) of a solid hemisphere = Curved Surface Area (CSA) + Base Area. So, CSA = TSA - Base Area. If TSA = \( 3 \pi r^2 \) and Base Area = \( \pi r^2 \), then CSA = \( 2 \pi r^2 \).

 

Question 20. If the radius of a sphere is half of the radius of another sphere, then their respective volumes are in the ratio
(1) 1:8
(2) 2:1
(3) 1:2
(4) 8:1
Answer: (1) 1:8
In simple words: If one sphere's radius is half of another's, it means the ratio of their radii is 1:2. Since the volume of a sphere depends on the cube of its radius, the ratio of their volumes will be the cube of the radii ratio, which is 1:8.

🎯 Exam Tip: When dealing with ratios of volumes for spheres, remember to cube the ratio of their radii. If \( r_1:r_2 = a:b \), then \( V_1:V_2 = a^3:b^3 \).

 

III. Answer the Following Questions

 

Question 1. Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq.cm and 110 cm respectively. Find its height and diameter.
Answer: Given that the circumference of the base of the cylinder is 110 cm and its curved surface area (CSA) is 4400 cm².
Circumference \( (2 \pi r) = 110 \) cm -------- (1)
Curved Surface Area \( (2 \pi r h) = 4400 \) cm² -------- (2)
Now, we can divide equation (2) by equation (1) to find the height:
\( \frac{2 \pi r h}{2 \pi r} = \frac{4400}{110} \)
\( \implies h = 40 \) cm
So, the height of the cylinder is 40 cm.
Next, we use equation (1) to find the radius:
\( 2 \times \frac{22}{7} \times r = 110 \)
\( \implies r = \frac{110 \times 7}{2 \times 22} = \frac{5 \times 7}{2} = \frac{35}{2} = 17.5 \) cm
We know that diameter \( d = 2 \times r \).
\( d = 2 \times \frac{35}{2} = 35 \) cm
Thus, the diameter of the circular cylinder is 35 cm. The height and diameter are both important dimensions.
In simple words: We are given the circumference of the base and the curved surface area of a cylinder. By dividing the curved surface area by the circumference, we found the height to be 40 cm. Then, using the circumference, we calculated the radius and finally the diameter, which is 35 cm.

🎯 Exam Tip: When both circumference and curved surface area are given for a cylinder, dividing CSA by circumference directly yields the height. This simplifies the calculation significantly.

 

Question 2. A mansion has 12 right cylindrical pillars each having radius 50 cm and height 3.5 m. Find the cost of painting the lateral surface of the pillars at Rs 20 per square metre.
Answer: Given that the radius of each cylindrical pillar \( (r) = 50 \) cm \( = 0.5 \) m.
The height of each pillar \( (h) = 3.5 \) m.
The curved surface area (lateral surface) of one pillar \( = 2 \pi r h \) sq. units.
\( = 2 \times \frac{22}{7} \times 0.5 \times 3.5 \) m²
\( = 2 \times \frac{22}{7} \times \frac{1}{2} \times \frac{35}{10} \) m²
\( = 22 \times \frac{5}{10} \) m² \( = 22 \times \frac{1}{2} = 11 \) m²
The curved surface area of 12 pillars \( = 12 \times 11 \) m² \( = 132 \) m².
The cost for painting the lateral surface is Rs 20 per square metre.
Total cost of painting \( = 132 \times \text{Rs } 20 = \text{Rs } 2640 \).
Painting pillars adds to the aesthetic appeal of the mansion.
In simple words: We first found the curved surface area of one cylindrical pillar, which is 11 m². Since there are 12 pillars, the total area to paint is 132 m². At a rate of Rs 20 per square metre, the total cost for painting all pillars is Rs 2640.

🎯 Exam Tip: Always ensure unit consistency (cm to m) before calculations. Remember that "lateral surface" refers to the curved surface area, excluding the top and bottom circles.

 

Question 3. The total surface area of a solid right circular cylinder is 231 cm². Its curved surface area is two thirds of the total surface area. Find the radius and height of the cylinder.
Answer: Given that the total surface area (TSA) of the cylinder \( = 231 \) cm².
Curved surface area (CSA) \( = \frac{2}{3} \times \text{TSA} \)
\( = \frac{2}{3} \times 231 = 2 \times 77 = 154 \) cm².
We know that the formula for CSA is \( 2 \pi r h = 154 \) cm² -------- (1)
The formula for TSA is \( 2 \pi r (h + r) = 231 \) cm².
\( \implies 2 \pi r h + 2 \pi r^2 = 231 \)
Substitute the value of \( 2 \pi r h \) from (1):
\( 154 + 2 \pi r^2 = 231 \)
\( \implies 2 \pi r^2 = 231 - 154 = 77 \)
\( \implies 2 \times \frac{22}{7} \times r^2 = 77 \)
\( \implies r^2 = \frac{77 \times 7}{2 \times 22} = \frac{7 \times 7}{2 \times 2} = \frac{49}{4} \)
\( \implies r = \sqrt{\frac{49}{4}} = \frac{7}{2} = 3.5 \) cm.
So, the radius of the cylinder is 3.5 cm.
Now, substitute the value of \( r \) into equation (1) to find the height:
\( 2 \pi r h = 154 \)
\( 2 \times \frac{22}{7} \times \frac{7}{2} \times h = 154 \)
\( \implies 22 \times h = 154 \)
\( \implies h = \frac{154}{22} = 7 \) cm.
Thus, the radius of the cylinder is 3.5 cm and its height is 7 cm. These dimensions help define the cylinder's shape.
In simple words: We used the given total surface area and the relationship that the curved surface area is two-thirds of the total. This helped us find the curved surface area. Then, we used the formulas for total and curved surface areas to solve for the radius, which is 3.5 cm, and the height, which is 7 cm.

🎯 Exam Tip: For cylinders, the relationship \( \text{TSA} = \text{CSA} + 2 \pi r^2 \) is very useful. If you know TSA and CSA, you can easily find \( 2 \pi r^2 \) and then the radius.

 

Question 4. The total surface area of a solid right circular cylinder is 1540 cm². If the height is four times the radius of the base, then find the height of the cylinder.
Answer: Let the radius of the cylinder be \( r \).
Given that the height of the cylinder \( (h) \) is four times the radius, so \( h = 4r \).
Total surface area \( (\text{TSA}) = 1540 \) cm².
The formula for TSA of a cylinder is \( 2 \pi r (h + r) \).
\( 2 \pi r (h + r) = 1540 \)
Substitute \( h = 4r \) into the formula:
\( 2 \pi r (4r + r) = 1540 \)
\( 2 \pi r (5r) = 1540 \)
\( 10 \pi r^2 = 1540 \)
\( 10 \times \frac{22}{7} \times r^2 = 1540 \)
\( r^2 = \frac{1540 \times 7}{10 \times 22} = \frac{154 \times 7}{22} \)
\( r^2 = 7 \times 7 = 49 \)
\( r = \sqrt{49} = 7 \) cm.
The radius of the cylinder is 7 cm.
Now, find the height using \( h = 4r \):
\( h = 4 \times 7 = 28 \) cm.
The height of the cylinder is 28 cm, which means it is a tall cylinder.
In simple words: We were given the total surface area and that the height is four times the radius. By putting this relationship into the total surface area formula, we solved for the radius, which is 7 cm. Then, we found the height by multiplying the radius by four, giving us 28 cm.

🎯 Exam Tip: When a relationship between radius and height is given (e.g., \( h = 4r \)), substitute it into the formula at the beginning to reduce the number of variables, making it easier to solve.

 

Question 5. If the vertical angle and the radius of a right circular cone are 60° and 15 cm respectively, then find its height and slant height.
Answer: Given that the radius \( (r) = 15 \) cm.
The vertical angle of the cone is 60°.
In a right circular cone, the semi-vertical angle \( (\alpha) \) is half of the vertical angle.
So, \( \alpha = \frac{60^\circ}{2} = 30^\circ \).
Consider the right-angled triangle formed by the radius, height, and slant height.
In this triangle, \( \tan(\alpha) = \frac{\text{radius}}{\text{height}} \).
\( \tan(30^\circ) = \frac{15}{h} \)
\( \implies \frac{1}{\sqrt{3}} = \frac{15}{h} \)
\( \implies h = 15 \sqrt{3} \) cm.
So, the height of the cone is \( 15 \sqrt{3} \) cm.
Now, to find the slant height \( (l) \), we can use \( \sin(\alpha) = \frac{\text{radius}}{\text{slant height}} \).
\( \sin(30^\circ) = \frac{15}{l} \)
\( \implies \frac{1}{2} = \frac{15}{l} \)
\( \implies l = 15 \times 2 = 30 \) cm.
Thus, the slant height of the cone is 30 cm. This helps us fully describe the cone's dimensions.
In simple words: We used the given radius and vertical angle to find the height and slant height of the cone. First, we found the semi-vertical angle (30°). Then, using trigonometry (tan for height and sin for slant height) in the right-angled triangle inside the cone, we calculated the height as \( 15 \sqrt{3} \) cm and the slant height as 30 cm.

🎯 Exam Tip: For cones, remember that the height, radius, and slant height form a right-angled triangle. Trigonometric ratios (sine, cosine, tangent) are very useful when angles are involved.

 

Question 6. The central angle and radius of a sector of a circular disc are 180° and 21 cm respectively. If the edges of the sector are joined together to make a hollow cone, then find the radius of the cone.
Answer: Given, Radius of the sector \( (r_s) = 21 \) cm.
The angle of the sector \( (\theta) = 180^\circ \).
When a sector is folded to form a cone, the radius of the sector becomes the slant height of the cone \( (l) \), and the arc length of the sector becomes the circumference of the base of the cone \( (2 \pi R_c) \), where \( R_c \) is the radius of the cone.
Let \( R_c \) be the radius of the cone.
Arc length of the sector \( = \frac{\theta}{360^\circ} \times 2 \pi r_s \)
\( = \frac{180^\circ}{360^\circ} \times 2 \pi \times 21 \)
\( = \frac{1}{2} \times 2 \pi \times 21 = 21 \pi \) cm.
Now, this arc length is equal to the circumference of the cone's base:
\( 2 \pi R_c = 21 \pi \)
\( \implies R_c = \frac{21 \pi}{2 \pi} = \frac{21}{2} = 10.5 \) cm.
So, the radius of the cone is 10.5 cm. This transformation from 2D to 3D is key.
In simple words: When a circular sector is folded into a cone, the arc of the sector becomes the circle at the cone's base, and the sector's radius becomes the cone's slant height. We calculated the length of the sector's arc and then used this length as the circumference of the cone's base to find the cone's radius, which is 10.5 cm.

🎯 Exam Tip: Remember that when a sector is rolled into a cone, the arc length of the sector is equal to the circumference of the base of the cone, and the radius of the sector is the slant height of the cone.

 

Question 7. If the curved surface area of a solid hemisphere is 2772 sq.cm, then find its total surface area.
Answer: Given that the curved surface area (CSA) of a solid hemisphere \( = 2772 \) cm².
The formula for the CSA of a hemisphere is \( 2 \pi r^2 \).
\( 2 \pi r^2 = 2772 \)
\( 2 \times \frac{22}{7} \times r^2 = 2772 \)
\( r^2 = \frac{2772 \times 7}{2 \times 22} = \frac{1386 \times 7}{22} = 63 \times 7 = 441 \)
\( r = \sqrt{441} = 21 \) cm.
So, the radius of the hemisphere is 21 cm.
The total surface area (TSA) of a solid hemisphere is \( 3 \pi r^2 \).
\( \text{TSA} = 3 \times \frac{22}{7} \times 21 \times 21 \)
\( = 3 \times 22 \times 3 \times 21 = 66 \times 63 = 4158 \) cm².
Thus, the total surface area of the solid hemisphere is 4158 cm². This area covers both the curved and flat parts.
*Aliter (Alternative method):*
We know that CSA of a hemisphere \( = 2 \pi r^2 \) and TSA of a solid hemisphere \( = 3 \pi r^2 \).
Therefore, TSA \( = \frac{3}{2} \times \text{CSA} \)
\( = \frac{3}{2} \times 2772 = 3 \times 1386 = 4158 \) cm².
In simple words: We were given the curved surface area of a hemisphere. We used this to find the radius of the hemisphere. Once we had the radius, we calculated the total surface area, which includes the flat base, and it came out to be 4158 cm². An easier way is to directly multiply the CSA by \( \frac{3}{2} \) to get the TSA.

🎯 Exam Tip: For a solid hemisphere, the Total Surface Area is always 1.5 times its Curved Surface Area (\( \text{TSA} = \frac{3}{2} \times \text{CSA} \)). This relationship can save time in calculations.

 

Question 8. An inner curved surface area of a hemispherical dome of a building needs to be painted. If the circumference of the base is 17.6 m, find the cost of painting it at the rate of Rs 5 per sq. m.
Answer: Given that the circumference of the base of the dome \( = 17.6 \) m.
The base of a hemispherical dome is a circle, so its circumference is \( 2 \pi r \).
\( 2 \pi r = 17.6 \)
\( 2 \times \frac{22}{7} \times r = 17.6 \)
\( r = \frac{17.6 \times 7}{2 \times 22} = \frac{0.8 \times 7}{2} = 0.4 \times 7 = 2.8 \) m.
So, the radius of the dome is 2.8 m.
The curved surface area of the dome (hemisphere) \( = 2 \pi r^2 \) sq. units.
\( = 2 \times \frac{22}{7} \times 2.8 \times 2.8 \) m²
\( = 2 \times \frac{22}{7} \times \frac{28}{10} \times \frac{28}{10} \) m²
\( = 2 \times 22 \times \frac{4}{10} \times \frac{28}{10} = \frac{44 \times 4 \times 28}{100} = \frac{4928}{100} = 49.28 \) m².
The cost of painting for one square metre \( = \text{Rs } 5 \).
Total cost of painting the curved surface \( = 49.28 \times \text{Rs } 5 = \text{Rs } 246.40 \).
Painting the dome enhances the building's appearance.
In simple words: We first found the radius of the hemispherical dome using the given circumference of its base. The radius is 2.8 m. Then, we calculated the curved surface area of the dome, which is 49.28 m². Finally, we multiplied this area by the painting rate of Rs 5 per square metre to get the total cost, which is Rs 246.40.

🎯 Exam Tip: For a hemispherical dome, the inner surface to be painted is its curved surface area, \( 2 \pi r^2 \). Use the circumference to find the radius first.

 

Question 9. Volume of a solid cylinder is 62.37 cu.cm. Find the radius if its height is 4.5 cm.
Answer: Given that the height of a cylinder \( (h) = 4.5 \) cm.
Volume of a solid cylinder \( = 62.37 \) cu. cm.
The formula for the volume of a cylinder is \( \pi r^2 h \).
\( \pi r^2 h = 62.37 \)
\( \frac{22}{7} \times r^2 \times 4.5 = 62.37 \)
\( r^2 = \frac{62.37 \times 7}{22 \times 4.5} \)
To simplify calculations, multiply numerator and denominator by 10 to remove decimals:
\( r^2 = \frac{623.7 \times 7}{22 \times 45} \)
\( r^2 = \frac{13.95 \times 7}{45} \) (since \( 623.7 / 22 = 28.35 \) and \( 28.35 / 4.5 = 6.3 \))
\( r^2 = 6.3 \times \frac{7}{45} \) (this path is also a bit tricky with decimals)
Let's try to simplify \( \frac{62.37}{4.5} \):
\( \frac{62.37}{4.5} = \frac{6237}{450} = \frac{138.6}{10} = 13.86 \)
So, \( r^2 = \frac{13.86 \times 7}{22} \)
\( r^2 = \frac{97.02}{22} = 4.41 \)
\( r = \sqrt{4.41} = 2.1 \) cm.
The radius of the cylinder is 2.1 cm.
In simple words: We used the given volume and height of the cylinder along with the formula for cylinder volume. By rearranging the formula, we solved for the radius squared. After calculating, we found that the radius of the cylinder is 2.1 cm.

🎯 Exam Tip: Be careful with decimal calculations. It's often helpful to convert decimals to fractions or multiply by powers of 10 to simplify the process, especially when dealing with \( \pi \).

 

Question 10. A rectangular sheet of metal foil with dimension 66 cm × 12 cm is rolled to form a cylinder of height 12 cm. Find the volume of the cylinder.
Answer:

When the rectangular sheet is rolled to form a cylinder:
The height of the cylinder \( (h) \) will be the shorter side of the rectangle, so \( h = 12 \) cm.
The base circumference of the cylinder will be the longer side of the rectangle, so Circumference \( = 66 \) cm.
We know Circumference \( = 2 \pi r \).
\( 2 \pi r = 66 \)
\( 2 \times \frac{22}{7} \times r = 66 \)
\( r = \frac{66 \times 7}{2 \times 22} = \frac{3 \times 7}{2} = \frac{21}{2} = 10.5 \) cm.
So, the radius of the cylinder is 10.5 cm.
Now, find the volume of the cylinder:
Volume \( (V) = \pi r^2 h \) cu. units.
\( V = \frac{22}{7} \times (10.5)^2 \times 12 \)
\( V = \frac{22}{7} \times 10.5 \times 10.5 \times 12 \)
\( V = 22 \times 1.5 \times 10.5 \times 12 \)
\( V = 33 \times 10.5 \times 12 \)
\( V = 346.5 \times 12 = 4158 \) cm³.
The volume of the cylinder formed is 4158 cm³. This shows how a flat sheet can hold volume when shaped.
In simple words: When a rectangular sheet is rolled into a cylinder, one side becomes the height and the other becomes the circumference of the base. We used the circumference (66 cm) to find the radius (10.5 cm) and the given height (12 cm) to calculate the volume of the cylinder, which is 4158 cm³.

🎯 Exam Tip: Understand which dimensions of the rectangle become which parts of the cylinder. The side that forms the height remains the height, and the other side forms the circumference of the base.

 

Question 11. The circumference of the base of a 12 m high wooden solid cone is 44 m. Find the volume.
Answer: Given that the height of the wooden solid cone \( (h) = 12 \) m.
Circumference of the base \( = 44 \) m.
The formula for circumference is \( 2 \pi r \).
\( 2 \pi r = 44 \)
\( 2 \times \frac{22}{7} \times r = 44 \)
\( r = \frac{44 \times 7}{2 \times 22} = 7 \) m.
So, the radius of the cone is 7 m.
Now, find the volume of the cone:
Volume \( (V) = \frac{1}{3} \pi r^2 h \) cu. units.
\( V = \frac{1}{3} \times \frac{22}{7} \times (7)^2 \times 12 \)
\( V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 12 \)
\( V = 22 \times 7 \times 4 \) (since \( \frac{12}{3} = 4 \) and one 7 cancels out)
\( V = 154 \times 4 = 616 \) m³.
The volume of the solid cone is 616 m³. This large volume indicates a substantial cone.
In simple words: We used the given base circumference of the cone to find its radius, which is 7 m. Then, with the radius and the given height of 12 m, we calculated the volume of the cone using its formula, resulting in 616 m³.

🎯 Exam Tip: Always calculate the radius from the circumference first before proceeding to find the volume or surface area. Remember the formula for cone volume: \( \frac{1}{3} \pi r^2 h \).

 

Question 12. Find the volume of the largest right circular cone that can be cut out of a cube whose edge is 14 cm.
Answer: Given that the edge of the cube \( = 14 \) cm.
For the largest right circular cone to be cut from a cube:
The diameter of the cone's base will be equal to the edge of the cube.
The height of the cone will also be equal to the edge of the cube.
So, diameter \( = 14 \) cm, which means radius \( (r) = \frac{14}{2} = 7 \) cm.
And height \( (h) = 14 \) cm.
Now, find the volume of the cone:
Volume \( (V) = \frac{1}{3} \pi r^2 h \) cu. units.
\( V = \frac{1}{3} \times \frac{22}{7} \times (7)^2 \times 14 \)
\( V = \frac{1}{3} \times \frac{22}{7} \times 7 \times 7 \times 14 \)
\( V = \frac{1}{3} \times 22 \times 7 \times 14 \)
\( V = \frac{1}{3} \times 154 \times 14 = \frac{2156}{3} \)
\( V = 718.67 \) cm³ (approximately).
The volume of the largest cone is about 718.67 cm³. This demonstrates packing efficiency.
In simple words: To get the largest cone from a cube, the cone's base diameter and height must both be equal to the cube's edge. So, with a cube edge of 14 cm, the cone has a radius of 7 cm and a height of 14 cm. We then calculated the cone's volume, which is approximately 718.67 cm³.

🎯 Exam Tip: For the "largest" shape cut from another, visualize how the dimensions relate. A cone fitting perfectly inside a cube will have its base diameter and height equal to the cube's side length.

 

Question 13. The thickness of a hemispherical bowl is 0.25 cm. The inner radius of the bowl is 5 cm. Find the outer curved surface area of the bowl. (Take π = \( \frac{22}{7} \))
Answer: Let \( r \) be the inner radius, \( R \) be the outer radius, and \( w \) be the thickness.
Given that the inner radius \( (r) = 5 \) cm.
Thickness \( (w) = 0.25 \) cm.
The outer radius \( (R) = r + w \)
\( R = 5 + 0.25 = 5.25 \) cm.
Now, calculate the outer curved surface area of the bowl (which is a hemisphere):
Outer Curved Surface Area \( = 2 \pi R^2 \)
\( = 2 \times \frac{22}{7} \times (5.25)^2 \)
\( = 2 \times \frac{22}{7} \times 5.25 \times 5.25 \)
\( = 44 \times 0.75 \times 5.25 \) (since \( \frac{5.25}{7} = 0.75 \))
\( = 33 \times 5.25 = 173.25 \) cm².
Therefore, the outer curved surface area of the hemispherical bowl is 173.25 cm². This area is vital for finishing or painting.
In simple words: First, we added the inner radius and the thickness to find the outer radius of the bowl. The outer radius is 5.25 cm. Then, we used the formula for the curved surface area of a hemisphere with this outer radius to calculate the answer, which is 173.25 cm².

🎯 Exam Tip: For hollow objects with thickness, distinguish between inner and outer dimensions. Always calculate the correct radius (inner or outer) before applying area or volume formulas.

 

Question 14. Volume of a hollow sphere is \( \frac{11352}{7} \) cm³. If the outer radius is 8 cm, find the inner radius of the sphere. (Take π = \( \frac{22}{7} \))
Answer:

Let \( R \) be the outer radius and \( r \) be the inner radius of the hollow sphere.
Given that the outer radius \( (R) = 8 \) cm.
Volume of the hollow sphere \( = \frac{11352}{7} \) cm³.
The formula for the volume of a hollow sphere is \( \frac{4}{3} \pi (R^3 - r^3) \).
\( \frac{4}{3} \pi (R^3 - r^3) = \frac{11352}{7} \)
Substitute \( \pi = \frac{22}{7} \) and \( R = 8 \):
\( \frac{4}{3} \times \frac{22}{7} (8^3 - r^3) = \frac{11352}{7} \)
\( \frac{88}{21} (512 - r^3) = \frac{11352}{7} \)
\( 512 - r^3 = \frac{11352}{7} \times \frac{21}{88} \)
\( 512 - r^3 = 11352 \times \frac{3}{88} \) (since \( \frac{21}{7} = 3 \))
\( 512 - r^3 = \frac{34056}{88} = 387 \)
\( 512 - r^3 = 387 \)
\( r^3 = 512 - 387 \)
\( r^3 = 125 \)
\( r = \sqrt[3]{125} = 5 \) cm.
Hence, the inner radius of the sphere is 5 cm. This reveals the hollow space inside.
In simple words: We used the formula for the volume of a hollow sphere, which involves the outer and inner radii. We plugged in the given volume and outer radius (8 cm) and then solved the equation for the inner radius. The inner radius was found to be 5 cm.

🎯 Exam Tip: The volume of a hollow sphere is the difference between the volume of the outer sphere and the inner sphere. Make sure to factor out \( \frac{4}{3} \pi \) correctly.

 

Question 15. How many litres of water will a hemispherical tank whose diameter is 4.2 m?
Answer: Given that the diameter of the hemispherical tank \( = 4.2 \) m.
Radius of the tank \( (r) = \frac{4.2}{2} = 2.1 \) m.
Volume of the hemisphere \( = \frac{2}{3} \pi r^3 \) cu. units.
\( = \frac{2}{3} \times \frac{22}{7} \times (2.1)^3 \)
\( = \frac{2}{3} \times \frac{22}{7} \times 2.1 \times 2.1 \times 2.1 \)
\( = \frac{2}{3} \times 22 \times 0.3 \times 2.1 \times 2.1 \) (since \( \frac{2.1}{7} = 0.3 \))
\( = 2 \times 22 \times 0.1 \times 2.1 \times 2.1 \) (since \( \frac{0.3}{3} = 0.1 \))
\( = 44 \times 0.1 \times 4.41 \)
\( = 4.4 \times 4.41 = 19.404 \) m³.
To convert cubic metres to litres, we use the conversion factor: \( 1 \) m³ \( = 1000 \) litres.
Volume in litres \( = 19.404 \times 1000 = 19404 \) litres.
The tank can hold 19,404 litres of water, which is a significant amount.
In simple words: We first found the radius from the given diameter of the hemispherical tank. Then, we used the formula for the volume of a hemisphere to calculate its volume in cubic metres, which is 19.404 m³. Finally, we converted this volume into litres by multiplying by 1000, giving us 19,404 litres.

🎯 Exam Tip: Remember the conversion: \( 1 \text{ m}^3 = 1000 \text{ litres} \). This is crucial for problems asking for capacity in litres. Always work with consistent units before converting.

 

Question 1. A vessel is in the form of a hollow hemisphere mounted by a hollow cylinder. The diameter of the hemisphere is 14 cm and the total height of the vessel is 13 cm. Find the inner surface area of the vessel.
Answer:

Given that the diameter of the hemisphere \( = 14 \) cm.
Radius of the hemisphere \( (r) = \frac{14}{2} = 7 \) cm.
The total height of the vessel \( = 13 \) cm.
Since the cylinder is mounted on the hemisphere, the height of the hemispherical part is equal to its radius, which is 7 cm.
Height of the cylindrical part \( (h_c) = \text{Total height} - \text{Radius of hemisphere} \)
\( h_c = 13 - 7 = 6 \) cm.
Radius of the cylindrical part \( (r_c) = \text{Radius of hemisphere} = 7 \) cm.
The inner surface area of the vessel will be the sum of the curved surface area of the cylinder and the curved surface area of the hemisphere.
Curved Surface Area of cylinder \( = 2 \pi r_c h_c \)
\( = 2 \times \frac{22}{7} \times 7 \times 6 \)
\( = 2 \times 22 \times 6 = 44 \times 6 = 264 \) cm².
Curved Surface Area of hemisphere \( = 2 \pi r^2 \)
\( = 2 \times \frac{22}{7} \times (7)^2 \)
\( = 2 \times \frac{22}{7} \times 7 \times 7 = 2 \times 22 \times 7 = 44 \times 7 = 308 \) cm².
Total inner surface area of the vessel \( = \text{CSA of cylinder} + \text{CSA of hemisphere} \)
\( = 264 + 308 = 572 \) cm².
The inner surface area of the vessel is 572 cm². This area is important for holding liquids or coating.
In simple words: The vessel is made of a cylinder on top of a hemisphere. We found the radius of both parts from the given diameter. Then we figured out the cylinder's height by subtracting the hemisphere's radius from the total height. We calculated the curved surface area of each part separately and added them up to get the total inner surface area, which is 572 cm².

🎯 Exam Tip: For combined solids, the total surface area is the sum of the visible surface areas of each component. Be careful to only include curved surfaces and exclude areas where the shapes join.

 

Question 2. A medicine capsule is in the shape of a cylinder with two hemispheres stuck to each of its ends. The length of the entire capsule is 14 mm and the diameter of the capsule is 5 mm. Find its surface area.
Answer:

Given that the diameter of the capsule \( = 5 \) mm.
Radius of the capsule \( (r) = \frac{5}{2} = 2.5 \) mm.
The total length of the capsule \( = 14 \) mm.
Since there are two hemispheres at the ends, the length contributed by each hemisphere is its radius. So, the total length from both hemispheres is \( 2r = 2 \times 2.5 = 5 \) mm.
Length of the cylindrical part \( (h_c) = \text{Total length of capsule} - \text{Length of two hemispheres} \)
\( h_c = 14 - 5 = 9 \) mm.
The surface area of the capsule will be the sum of the curved surface area of the cylinder and the curved surface areas of the two hemispheres.
Curved Surface Area of cylinder \( = 2 \pi r h_c \)
\( = 2 \times \frac{22}{7} \times 2.5 \times 9 \)
\( = 2 \times \frac{22}{7} \times \frac{5}{2} \times 9 \)
\( = \frac{22 \times 5 \times 9}{7} = \frac{990}{7} \) mm².
Curved Surface Area of one hemisphere \( = 2 \pi r^2 \).
Curved Surface Area of two hemispheres \( = 2 \times 2 \pi r^2 = 4 \pi r^2 \)
\( = 4 \times \frac{22}{7} \times (2.5)^2 \)
\( = 4 \times \frac{22}{7} \times \frac{5}{2} \times \frac{5}{2} \)
\( = \frac{22 \times 5 \times 5}{7} = \frac{550}{7} \) mm².
Total surface area of the capsule \( = \text{CSA of cylinder} + \text{CSA of two hemispheres} \)
\( = \frac{990}{7} + \frac{550}{7} = \frac{990 + 550}{7} = \frac{1540}{7} = 220 \) mm².
The surface area of the medicine capsule is 220 mm². This area is important for drug coating.
In simple words: We first found the radius from the diameter. Then, we calculated the length of the cylindrical part by subtracting the total length of the two hemispheres from the capsule's total length. Finally, we added the curved surface areas of the cylinder and the two hemispheres to get the total surface area of the capsule, which is 220 mm².

🎯 Exam Tip: For objects composed of multiple shapes, carefully identify the individual shapes and their dimensions. Ensure you only include the exposed surface areas in your final calculation for the combined solid.

 

Question 3. From a solid cylinder whose height is 2.4 cm and diameter 1.4 cm, a conical cavity of the same height and same diameter is hollowed out. Find the total surface area of the remaining solid to the nearest cm².
Answer:

Given that the height of the solid cylinder \( (h) = 2.4 \) cm.
Diameter of the cylinder \( (d) = 1.4 \) cm.
Radius of the cylinder \( (r) = \frac{1.4}{2} = 0.7 \) cm.
A conical cavity of the same height and diameter is hollowed out.
For the conical cavity: height \( (h_c) = 2.4 \) cm, radius \( (r_c) = 0.7 \) cm.
To find the total surface area of the remaining solid, we need:
1. Curved surface area of the cylinder (outer part)
2. Area of the base of the cylinder
3. Curved surface area of the cone (inner part)
First, calculate the slant height \( (l) \) of the conical cavity:
\( l = \sqrt{r_c^2 + h_c^2} = \sqrt{(0.7)^2 + (2.4)^2} \)
\( l = \sqrt{0.49 + 5.76} = \sqrt{6.25} = 2.5 \) cm.
Curved Surface Area of cylinder \( = 2 \pi r h \)
\( = 2 \times \frac{22}{7} \times 0.7 \times 2.4 \)
\( = 2 \times 22 \times 0.1 \times 2.4 = 44 \times 0.24 = 10.56 \) cm².
Area of the base of the cylinder \( = \pi r^2 \)
\( = \frac{22}{7} \times (0.7)^2 = \frac{22}{7} \times 0.7 \times 0.7 \)
\( = 22 \times 0.1 \times 0.7 = 2.2 \times 0.7 = 1.54 \) cm².
Curved Surface Area of cone \( = \pi r_c l \)
\( = \frac{22}{7} \times 0.7 \times 2.5 \)
\( = 22 \times 0.1 \times 2.5 = 2.2 \times 2.5 = 5.5 \) cm².
Total surface area of the remaining solid \( = \text{CSA of cylinder} + \text{Base area of cylinder} + \text{CSA of cone} \)
\( = 10.56 + 1.54 + 5.5 \)
\( = 12.1 + 5.5 = 17.6 \) cm².
The total surface area of the remaining solid is 17.6 cm². This composite shape now has a new exposed internal surface.
In simple words: We calculated the curved surface area of the cylinder, the area of its base, and the curved surface area of the conical cavity. To do this, we first needed to find the slant height of the cone. After adding all these areas together, the total surface area of the remaining solid is 17.6 cm².

🎯 Exam Tip: For solids with hollowed-out parts, the total surface area of the remaining solid includes the outer surfaces plus the inner surface created by the cavity. Remember to find the slant height for conical parts.

 

Question 4. A 20 m deep well with diameter 7 m is dug and the earth from digging is evenly spread out to form a platform 22 m by 14 m. Find the height of the platform.
Answer: Given that the diameter of the cylindrical well \( = 7 \) m.
Radius of the cylinder \( (r) = \frac{7}{2} \) m.
Depth of the well \( (h) = 20 \) m.
Volume of the earth taken out from the well \( = \pi r^2 h \)
\( = \frac{22}{7} \times (\frac{7}{2})^2 \times 20 \)
\( = \frac{22}{7} \times \frac{7}{2} \times \frac{7}{2} \times 20 \)
\( = 11 \times 7 \times 10 = 770 \) m³.
This volume of earth is spread out to form a cuboidal platform.
Given dimensions of the platform: length \( (l_p) = 22 \) m, breadth \( (b_p) = 14 \) m.
Let the height of the platform be \( (h_p) \).
Volume of the platform \( = l_p \times b_p \times h_p \)
\( = 22 \times 14 \times h_p \) m³.
Since the volume of the earth taken out is equal to the volume of the platform:
\( 22 \times 14 \times h_p = 770 \)
\( h_p = \frac{770}{22 \times 14} = \frac{770}{308} \)
\( h_p = 2.5 \) m.
Thus, the required height of the platform is 2.5 m. This height is essential for its stability.
In simple words: First, we found the volume of earth dug out from the cylindrical well. This volume is 770 m³. Then, we used this volume and the given length and breadth of the platform to calculate its height. We found the height of the platform to be 2.5 m.

🎯 Exam Tip: When earth is dug out and spread to form another shape, the volume of the dug-out earth is equal to the volume of the new structure. Ensure units are consistent throughout the calculation.

 

Question 5. The perimeters of the ends of the frustum of a cone are 207.24 cm and 169.56 cm. If the height of the frustum is 8 cm, find the whole surface area of the frustum. [Use π = 3.14]
Answer:

Let \( R \) be the radius of the top end and \( r \) be the radius of the bottom end.
Given that \( \pi = 3.14 \).
Perimeter of the top end \( = 207.24 \) cm.
\( 2 \pi R = 207.24 \)
\( 2 \times 3.14 \times R = 207.24 \)
\( R = \frac{207.24}{2 \times 3.14} = \frac{207.24}{6.28} = 33 \) cm.
Perimeter of the bottom end \( = 169.56 \) cm.
\( 2 \pi r = 169.56 \)
\( 2 \times 3.14 \times r = 169.56 \)
\( r = \frac{169.56}{2 \times 3.14} = \frac{169.56}{6.28} = 27 \) cm.
Height of the frustum \( (h) = 8 \) cm.
First, find the slant height \( (l) \) of the frustum:
\( l = \sqrt{h^2 + (R - r)^2} \)
\( l = \sqrt{8^2 + (33 - 27)^2} \)
\( l = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10 \) cm.
The whole surface area of the frustum \( = \pi (R^2 + r^2) + \pi (R + r) l \)
\( = \pi [R^2 + r^2 + (R + r) l] \)
\( = 3.14 [33^2 + 27^2 + (33 + 27) \times 10] \)
\( = 3.14 [1089 + 729 + (60) \times 10] \)
\( = 3.14 [1089 + 729 + 600] \)
\( = 3.14 [2418] \)
\( = 7592.52 \) cm².
The whole surface area of the frustum is 7592.52 cm². This area is large due to its size.
In simple words: We used the perimeters of the frustum's top and bottom to find their respective radii, which are 33 cm and 27 cm. Then, using the height (8 cm) and the radii, we calculated the slant height (10 cm). Finally, we applied the formula for the whole surface area of a frustum, adding all parts together, to get 7592.52 cm².

🎯 Exam Tip: For a frustum, calculate the radii from the perimeters first. Remember the formula for slant height \( l = \sqrt{h^2 + (R - r)^2} \) and the total surface area formula, which includes the areas of both circular bases and the curved surface area.

 

Question 6. A cuboid-shaped slab of iron whose dimensions are 55 cm × 40 cm × 15 cm is melted and recast into a pipe. The outer diameter and thickness of the pipe are 8 cm and 1 cm respectively. Find the length of the pipe. (Take π = \( \frac{22}{7} \))
Answer:

For the iron slab (cuboid):
Length \( (l) = 55 \) cm, Breadth \( (b) = 40 \) cm, Height \( (h) = 15 \) cm.
Volume of the iron slab \( = l \times b \times h \)
\( = 55 \times 40 \times 15 = 2200 \times 15 = 33000 \) cm³.
For the iron pipe (hollow cylinder):
Outer diameter \( = 8 \) cm, so Outer radius \( (R) = \frac{8}{2} = 4 \) cm.
Thickness \( (w) = 1 \) cm.
Inner radius \( (r) = R - w = 4 - 1 = 3 \) cm.
Let the length of the pipe be \( (L) \).
The volume of the iron pipe \( = \pi (R^2 - r^2) L \)
\( = \frac{22}{7} ((4)^2 - (3)^2) L \)
\( = \frac{22}{7} (16 - 9) L = \frac{22}{7} (7) L = 22 L \) cm³.
Since the iron slab is melted and recast into the pipe, their volumes must be equal.
Volume of iron pipe \( = \) Volume of iron slab
\( 22 L = 33000 \)
\( L = \frac{33000}{22} = 1500 \) cm.
Therefore, the length of the pipe is 1500 cm or 15 m. This length is substantial for a pipe.
In simple words: First, we calculated the volume of the iron slab. Then, for the pipe, we used the outer diameter and thickness to find its inner and outer radii. We set the volume of the iron slab equal to the formula for the volume of a hollow cylinder (the pipe) and solved for the length of the pipe. The length of the pipe is 1500 cm.

🎯 Exam Tip: When a solid is melted and recast, its volume remains constant. For hollow cylinders (pipes), remember to use the difference of the squares of the outer and inner radii in the volume formula: \( \pi (R^2 - r^2) L \).

 

Question 7. The perimeter of the ends of a frustum of a cone are 44 cm and 8.4π cm. If the depth is 14 cm., then find its volume.
Answer: Let the radius of the top end be \( R \) and the radius of the bottom end be \( r \).
Given the depth (height) of the frustum \( (h) = 14 \) cm.
Perimeter of the top end \( = 44 \) cm.
\( 2 \pi R = 44 \)
\( 2 \times \frac{22}{7} \times R = 44 \)
\( R = \frac{44 \times 7}{2 \times 22} = 7 \) cm.
Perimeter of the bottom end \( = 8.4 \pi \) cm.
\( 2 \pi r = 8.4 \pi \)
\( r = \frac{8.4 \pi}{2 \pi} = \frac{8.4}{2} = 4.2 \) cm.
Now, find the volume of the frustum:
Volume \( (V) = \frac{1}{3} \pi h (R^2 + r^2 + Rr) \)
\( V = \frac{1}{3} \times \frac{22}{7} \times 14 (7^2 + (4.2)^2 + 7 \times 4.2) \)
\( V = \frac{1}{3} \times 22 \times 2 (49 + 17.64 + 29.4) \)
\( V = \frac{44}{3} (49 + 17.64 + 29.4) \)
\( V = \frac{44}{3} (96.04) \)
\( V = \frac{4225.76}{3} \)
\( V = 1408.586... \) cm³.
\( V \approx 1408.6 \) cm³.
The volume of the frustum is approximately 1408.6 cm³. This calculation is crucial for container capacity.
In simple words: We used the perimeters of the frustum's ends to find the radii of the top (7 cm) and bottom (4.2 cm). With these radii and the given height (14 cm), we applied the volume formula for a frustum of a cone. The calculated volume is approximately 1408.6 cm³.

🎯 Exam Tip: The volume of a frustum formula involves both radii and the height. Remember the formula \( V = \frac{1}{3} \pi h (R^2 + r^2 + Rr) \) and derive R and r from the given perimeters.

 

Question 8. A tent is in the shape of a right circular cylinder surmounted by a cone. The total height and the diameter of the base are 13.5 m and 28 m. If the height of the cylindrical portion is 3 m, find the total surface area of the tent.
Answer:

Given that the total height of the tent \( = 13.5 \) m.
Diameter of the base \( = 28 \) m.
Radius of the base \( (r) = \frac{28}{2} = 14 \) m.
Height of the cylindrical portion \( (h_{cyl}) = 3 \) m.
Height of the conical portion \( (h_{cone}) = \text{Total height} - h_{cyl} \)
\( = 13.5 - 3 = 10.5 \) m.
The total surface area of the tent includes:
1. Curved surface area of the cylindrical portion
2. Curved surface area of the conical portion
3. Area of the base of the cylindrical portion (floor of the tent)
First, calculate the slant height \( (l) \) of the cone:
\( l = \sqrt{r^2 + h_{cone}^2} = \sqrt{14^2 + (10.5)^2} \)
\( l = \sqrt{196 + 110.25} = \sqrt{306.25} = 17.5 \) m.
Curved Surface Area of cylinder \( = 2 \pi r h_{cyl} \)
\( = 2 \times \frac{22}{7} \times 14 \times 3 \)
\( = 2 \times 22 \times 2 \times 3 = 264 \) m².
Curved Surface Area of cone \( = \pi r l \)
\( = \frac{22}{7} \times 14 \times 17.5 \)
\( = 22 \times 2 \times 17.5 = 44 \times 17.5 = 770 \) m².
Area of the base (floor) of the tent \( = \pi r^2 \)
\( = \frac{22}{7} \times 14^2 = \frac{22}{7} \times 14 \times 14 \)
\( = 22 \times 2 \times 14 = 44 \times 14 = 616 \) m².
Total surface area of the tent \( = \text{CSA of cylinder} + \text{CSA of cone} + \text{Base Area} \)
\( = 264 + 770 + 616 = 1650 \) m².
The total surface area of the tent is 1650 m². This area dictates the amount of fabric needed.
In simple words: The tent is a cylinder with a cone on top. We found the radius from the diameter and the cone's height by subtracting the cylinder's height from the total height. We calculated the cone's slant height. Then, we found the curved surface area of the cylinder, the curved surface area of the cone, and the area of the tent's floor, and added them all together to get the total surface area of 1650 m².

🎯 Exam Tip: For composite solids like a tent, identify all exposed surfaces. The total surface area is the sum of the curved surface areas of the cylinder and cone, plus the base area if it's considered part of the tent's material.

I. Multiple Choice Questions:

 

Question 1. The curved surface area of a right circular cylinder of radius 1 cm and height 1 cm is equal to
(1) \( \pi \text{ cm}^2 \)
(2) \( 2\pi \text{ cm}^2 \)
(3) \( 3\pi \text{ cm}^2 \)
(4) \( 2 \text{ cm}^2 \)
Answer: (2) \( 2\pi \text{ cm}^2 \)
In simple words: To find the curved surface area of a cylinder, we use the formula \( 2\pi rh \). When the radius and height are both 1 cm, the area becomes \( 2\pi \times 1 \times 1 \), which is \( 2\pi \text{ cm}^2 \). This formula measures the side surface, not the top or bottom circles.

🎯 Exam Tip: Remember the curved surface area formula for a cylinder: \( 2\pi rh \). Pay close attention to the units and substitute values carefully.

 

Question 2. The total surface area of a solid right circular cylinder whose radius is half of its height h is equal to sq. units.
(1) \( \frac{3}{2} \pi h \)
(2) \( \frac{2}{3} \pi h^{2} \)
(3) \( \frac{3}{2} \pi h^{2} \)
(4) \( \frac{2}{3} \pi h \)
Answer: (3) \( \frac{3}{2} \pi h^{2} \)
In simple words: The total surface area of a cylinder is found by adding the area of its two circular bases and its curved side. If the radius \( r \) is half of the height \( h \) (so \( r = h/2 \)), we can put this into the total surface area formula, \( 2\pi r(h + r) \). This simplifies to \( \frac{3}{2} \pi h^2 \), showing how the area depends on the height when radius is related to height.

🎯 Exam Tip: When given a relationship between radius and height (like \( r = h/2 \)), always substitute it into the formula for the desired quantity (like TSA) to express the answer in terms of a single variable.

 

Question 3. Base area of a right circular cylinder is 80 cm². If its height is 5 cm, then the volume is equal to
(1) 400 cm³
(2) 16 cm³
(3) 200 cm³
Answer: (1) 400 cm³
In simple words: The volume of a cylinder is found by multiplying the area of its base by its height. Since the base area is 80 cm² and the height is 5 cm, the total volume is \( 80 \times 5 = 400 \text{ cm}^3 \). This is a straightforward application of the volume formula for cylinders.

🎯 Exam Tip: Remember that the volume of any prism or cylinder is simply (Base Area) × (Height). This makes calculations easier when the base area is already given.

 

Question 4. If the total surface area of a solid right circular cylinder is 200\( \pi \) cm² and its radius is 5 cm, then the sum of its height and radius is
(1) 20 cm
(2) 25 cm
(3) 30 cm
(4) 15 cm
Answer: (1) 20 cm
In simple words: The total surface area of a cylinder is given by \( 2\pi r(h + r) \). If we know the total surface area is 200\( \pi \) cm² and the radius \( r \) is 5 cm, we can substitute these values into the formula. This lets us easily find the sum of the height and radius \( (h+r) \) by dividing the total surface area by \( 2\pi r \).

🎯 Exam Tip: Be careful with the units and whether the given area includes \( \pi \) or not. If \( \pi \) is already in the given area, it often simplifies out during calculation.

 

Question 5. The curved surface area of a right circular cylinder whose radius is a units and height is b units, is equal to
(1) \( \pi a^2 b \text{ sq.cm} \)
(2) \( 2\pi ab \text{ sq.cm} \)
(3) \( 2\pi \text{ sq.cm} \)
(4) \( 2 \text{ sq.cm} \)
Answer: (2) \( 2\pi ab \text{ sq.cm} \)
In simple words: The formula for the curved surface area of a cylinder is \( 2\pi rh \), where \( r \) is the radius and \( h \) is the height. If the radius is \( a \) and the height is \( b \), we just substitute these letters into the formula to get \( 2\pi ab \). This area represents only the side surface of the cylinder.

🎯 Exam Tip: Memorize the basic formulas for curved surface area and total surface area for common shapes. Understanding which variables represent radius, height, or slant height is crucial.

 

Question 6. Radius and height of a right circular cone and that of a right circular cylinder are respectively, equal. If the volume of the cylinder is 120 cm³, then the volume of the cone is equal to
(1) 1200 cm³
(2) 360 cm³
(3) 40 cm³
(4) 90 cm³
Answer: (3) 40 cm³
In simple words: The volume of a cone is always one-third the volume of a cylinder if they have the same base radius and the same height. So, if the cylinder's volume is 120 cm³, the cone's volume will be \( \frac{1}{3} \times 120 = 40 \text{ cm}^3 \). This is a key relationship between these two geometric shapes.

🎯 Exam Tip: Remember the fundamental relationship: \( \text{Volume of cone} = \frac{1}{3} \times \text{Volume of cylinder (with same base and height)} \). This simplifies calculations when one volume is known.

 

Question 7. If the diameter and height of a right circular cone are 12 cm and 8 cm respectively, then the slant height is
(1) 10 cm
(2) 20 cm
(3) 30 cm
(4) 96 cm
Answer: (1) 10 cm
In simple words: For a right circular cone, the radius, height, and slant height form a right-angled triangle. We can use the Pythagorean theorem, \( l^2 = r^2 + h^2 \), to find the slant height \( l \). Given a diameter of 12 cm (so radius \( r = 6 \) cm) and a height \( h = 8 \) cm, the slant height is \( \sqrt{6^2 + 8^2} = \sqrt{36+64} = \sqrt{100} = 10 \text{ cm} \). This is a classic application of the Pythagorean triple (6, 8, 10).

🎯 Exam Tip: Always convert diameter to radius first (\( r = d/2 \)) before applying formulas. Recognize Pythagorean triples (like 3,4,5 or 6,8,10) to quickly solve for missing sides in right-angled triangles.

 

Question 8. If the circumference at the base of a right circular cone and the slant height are 120\( \pi \) cm and 10 cm respectively, then the curved surface area of the cone is equal to
(1) 1200\( \pi \) cm²
(2) 600\( \pi \) cm²
(3) 300\( \pi \) cm²
(4) 600 cm²
Answer: (2) 600\( \pi \) cm²
In simple words: The curved surface area of a cone is found using \( \pi rl \). We are given the circumference \( 2\pi r = 120\pi \) cm, which tells us that the radius \( r = 60 \) cm. With the slant height \( l = 10 \) cm, we can calculate the curved surface area as \( \pi \times 60 \times 10 = 600\pi \text{ cm}^2 \). Knowing the circumference helps us find the radius needed for the area formula.

🎯 Exam Tip: The curved surface area of a cone can be calculated directly if you know the circumference of the base and the slant height: \( \text{CSA} = \frac{1}{2} \times \text{Circumference} \times \text{Slant height} \).

 

Question 9. If the volume and the base area of a right circular cone are 48\( \pi \) cm³ and 12\( \pi \) cm² respectively, then the height of the cone is equal to
(1) 6 cm
(2) 8 cm
(3) 10 cm
(4) 12 cm
Answer: (4) 12 cm
In simple words: The volume of a cone is given by \( \frac{1}{3} \times \text{base area} \times \text{height} \). If we know the volume (48\( \pi \) cm³) and the base area (12\( \pi \) cm²), we can easily find the height by rearranging the formula. This gives us \( 48\pi = \frac{1}{3} \times 12\pi \times h \), which simplifies to \( h = 12 \text{ cm} \).

🎯 Exam Tip: The volume formula for a cone (\( V = \frac{1}{3} \pi r^2 h \)) can be rewritten as \( V = \frac{1}{3} \times (\text{Base Area}) \times h \). This form is useful when the base area is directly provided.

 

Question 10. If the height and the base area of a right circular cone are 5 cm and 48 sq.cm respectively, then the volume of the cone is equal to
(1) 240 cm³
(2) 120 cm³
(3) 80 cm³
(4) 480 cm³
Answer: (3) 80 cm³
In simple words: To calculate the volume of a cone, you multiply one-third of the base area by its height. Here, the base area is 48 sq.cm and the height is 5 cm, so the volume is \( \frac{1}{3} \times 48 \times 5 = 16 \times 5 = 80 \text{ cm}^3 \). This formula helps us find the space a cone occupies.

🎯 Exam Tip: Double-check whether you are given the radius or the base area directly. If the base area is given, simply use it in the volume formula: \( V = \frac{1}{3} \times \text{Base Area} \times h \).

 

Question 11. The ratios of the respective heights and the respective radii of two cylinders are 1 : 2 and 2 : 1 respectively. Then their respective volumes are in the ratio
(1) 4:1
(3) 2:1
(4) 1:2
Answer: (3) 2:1
In simple words: The volume of a cylinder is proportional to \( r^2 h \). If the ratio of heights (\( h_1:h_2 \)) is 1:2 and the ratio of radii (\( r_1:r_2 \)) is 2:1, then the ratio of their volumes is \( r_1^2 h_1 : r_2^2 h_2 \). Substituting the ratios, we get \( (2)^2 \times 1 : (1)^2 \times 2 = 4 \times 1 : 1 \times 2 = 4:2 \), which simplifies to 2:1. This shows how changes in dimensions affect the overall volume.

🎯 Exam Tip: For ratios of volumes or areas, express the formula in terms of \( r \) and \( h \), then substitute the given ratios directly. Remember to square the radius when calculating volume ratios for cylinders.

 

Question 12. If the radius of a sphere is 2 cm, then the curved surface area of the sphere is equal to
(1) 8\( \pi \) cm²
(2) 16 cm²
(3) 12\( \pi \) cm²
(4) 16\( \pi \) cm²
Answer: (4) 16\( \pi \) cm²
In simple words: The curved surface area of a sphere is given by the formula \( 4\pi r^2 \). If the radius \( r \) is 2 cm, we substitute this value into the formula to get \( 4\pi (2)^2 = 4\pi \times 4 = 16\pi \text{ cm}^2 \). This area is the entire outer surface of the ball-shaped object.

🎯 Exam Tip: The curved surface area and total surface area of a sphere are the same. Make sure to square the radius correctly in the formula \( 4\pi r^2 \).

 

Question 13. The total surface area of a solid hemisphere of diameter 2 cm is equal to
(2) 12\( \pi \) cm²
(3) 4\( \pi \) cm²
(4) 3\( \pi \) cm²
Answer: (4) 3\( \pi \) cm²
In simple words: A solid hemisphere has a curved surface and a flat circular base. The radius is 1 cm (since diameter is 2 cm). Its total surface area is \( 2\pi r^2 \) (curved part) + \( \pi r^2 \) (flat base) = \( 3\pi r^2 \). So, for \( r = 1 \) cm, the area is \( 3\pi (1)^2 = 3\pi \text{ cm}^2 \). This calculation includes both the rounded top and the flat bottom of the half-sphere.

🎯 Exam Tip: For a solid hemisphere, total surface area is \( 3\pi r^2 \). Remember to include the flat base area (\( \pi r^2 \)) in addition to the curved surface area (\( 2\pi r^2 \)).

 

Question 14. If the volume of a sphere is \( \frac{9}{16} \pi \) cu.cm, then its radius is
(1) \( \frac{4}{3} \) cm
(2) \( \frac{3}{4} \) cm
(3) \( \frac{3}{2} \) cm
(4) \( \frac{2}{3} \) cm
Answer: (2) \( \frac{3}{4} \) cm
In simple words: The formula for the volume of a sphere is \( \frac{4}{3} \pi r^3 \). If we set this equal to the given volume of \( \frac{9}{16} \pi \), we can solve for the radius \( r \). By cancelling \( \pi \) and rearranging, we find \( r^3 = \frac{27}{64} \), which means \( r = \frac{3}{4} \text{ cm} \). Finding the cube root is important here.

🎯 Exam Tip: Be careful with algebraic manipulation when solving for \( r^3 \). Always take the cube root of both the numerator and the denominator of the fraction to get the final radius.

 

Question 15. The surface areas of two spheres are in the ratio of 9 : 25. Then their volumes are in the ratio
(1) 81: 625
(2) 729: 15625
(4) 27: 125
Answer: (4) 27: 125
In simple words: The surface area of a sphere is proportional to \( r^2 \), and its volume is proportional to \( r^3 \). If the ratio of surface areas (\( r_1^2 : r_2^2 \)) is 9:25, then the ratio of their radii (\( r_1:r_2 \)) is \( \sqrt{9}:\sqrt{25} = 3:5 \). Therefore, the ratio of their volumes (\( r_1^3 : r_2^3 \)) will be \( 3^3:5^3 = 27:125 \). This shows how the scaling of dimensions affects area and volume differently.

🎯 Exam Tip: For problems involving ratios of areas and volumes, remember that Area \( \propto r^2 \) and Volume \( \propto r^3 \). First find the ratio of radii from the area ratio, then cube it for the volume ratio.

 

Question 16. The total surface area of a solid hemisphere whose radius is a units, is equal to
(1) \( 2\pi a^2 \) sq. units
(2) \( 3\pi a^2 \) sq. units
(3) \( 3\pi a \) sq. units
(4) \( 3a^2 \) sq. units
Answer: (2) \( 3\pi a^2 \) sq. units
In simple words: For a solid hemisphere, the total surface area includes both the curved part and the flat circular base. The curved surface area is \( 2\pi r^2 \), and the base area is \( \pi r^2 \). Adding them gives \( 3\pi r^2 \). If the radius is \( a \), then the total surface area is \( 3\pi a^2 \). This sum accounts for all exposed surfaces of the half-sphere.

🎯 Exam Tip: Differentiate between the curved surface area (\( 2\pi r^2 \)) and the total surface area (\( 3\pi r^2 \)) for a solid hemisphere. The "solid" implies the base is also part of the surface.

 

Question 17. If the surface area of a sphere is 100\( \pi \) cm², then its radius is equal to
(1) 25 cm
(2) 100 cm
(3) 5 cm
(4) 10 cm
Answer: (3) 5 cm
In simple words: The surface area of a sphere is given by \( 4\pi r^2 \). If this area is 100\( \pi \) cm², we can set up the equation \( 4\pi r^2 = 100\pi \). By cancelling \( \pi \) and dividing by 4, we find \( r^2 = 25 \), which means the radius \( r = 5 \text{ cm} \). This calculation reverses the surface area formula to find the sphere's size.

🎯 Exam Tip: When \( \pi \) appears on both sides of an equation in geometry problems, you can often cancel it out, simplifying the calculation for the unknown variable.

 

Question 18. If the surface area of a sphere is 36\( \pi \) cm², then the volume of the sphere is equal to
(1) 12\( \pi \) cm³
(2) 36\( \pi \) cm³
(3) 72\( \pi \) cm³
(4) 108\( \pi \) cm³
Answer: (2) 36\( \pi \) cm³
In simple words: First, we find the radius from the given surface area (\( 4\pi r^2 = 36\pi \)). This gives \( r^2 = 9 \), so \( r = 3 \) cm. Then, we use this radius to find the volume of the sphere (\( V = \frac{4}{3} \pi r^3 \)). Substituting \( r = 3 \), we get \( V = \frac{4}{3} \pi (3)^3 = \frac{4}{3} \pi \times 27 = 4\pi \times 9 = 36\pi \text{ cm}^3 \). This is a two-step problem where the radius is the link between surface area and volume.

🎯 Exam Tip: For problems that link surface area and volume, always find the radius first, as it is common to both formulas. Then use the radius to calculate the other requested quantity.

 

Question 19. If the total surface area of a solid hemisphere is 12\( \pi \) cm² then its curved surface area is equal to
(1) 6\( \pi \) cm²
(2) 24\( \pi \) cm²
(3) 36\( \pi \) cm²
(4) 8\( \pi \) cm²
Answer: (4) 8\( \pi \) cm²
In simple words: The total surface area of a solid hemisphere is \( 3\pi r^2 \). Given that this is 12\( \pi \) cm², we can find the radius \( r \): \( 3\pi r^2 = 12\pi \implies r^2 = 4 \implies r = 2 \) cm. The curved surface area of a hemisphere is \( 2\pi r^2 \). Using \( r = 2 \) cm, the curved surface area is \( 2\pi (2)^2 = 2\pi \times 4 = 8\pi \text{ cm}^2 \). It's important to know the difference between total and curved surface area.

🎯 Exam Tip: Always identify if the question asks for the total surface area (including the base) or just the curved surface area. For a solid hemisphere, TSA is \( 3\pi r^2 \) and CSA is \( 2\pi r^2 \).

 

Question 20. If the radius of a sphere is half of the radius of another sphere, then their respective volumes are in the ratio
(1) 1:8
(2) 2:1
(3) 1:2
(4) 8:1
Answer: (1) 1:8
In simple words: The volume of a sphere is proportional to the cube of its radius (\( r^3 \)). If one sphere's radius (\( r_1 \)) is half of another's (\( r_2 \)), then \( r_1:r_2 = 1:2 \). Therefore, the ratio of their volumes (\( V_1:V_2 \)) will be \( r_1^3:r_2^3 = 1^3:2^3 = 1:8 \). This shows how a small change in radius makes a much bigger change in volume.

🎯 Exam Tip: When dealing with ratios of volumes, remember that if linear dimensions are in ratio \( a:b \), then volumes are in ratio \( a^3:b^3 \). This is a fundamental scaling principle in geometry.

 

Question 1. Curved surface area and circumference at the base of a solid right circular cylinder are 4400 sq.cm and 110 cm respectively. Find its height and diameter.
Answer: Given, the circumference of the base of a cylinder is 110 cm.
\( 2\pi r = 110 \) ....... (1)
The curved surface area is 4400 cm².
\( 2\pi rh = 4400 \) ....... (2)
To find the height, divide equation (2) by equation (1):
\( \frac{2 \pi r h}{2 \pi r} = \frac{4400}{110} \)
\( \implies h = 40 \text{ cm} \)
So, the height of the cylinder is 40 cm.
Now, from equation (1), find the radius:
\( 2\pi r = 110 \)
\( 2 \times \frac{22}{7} \times r = 110 \)
\( \implies r = \frac{110 \times 7}{2 \times 22} = \frac{5 \times 7}{2} = \frac{35}{2} = 17.5 \text{ cm} \)
We know that the diameter \( d = 2 \times \text{radius} \).
\( d = 2 \times 17.5 = 35 \text{ cm} \)
Thus, the diameter of the circular cylinder is 35 cm.
In simple words: We are given how big the bottom edge of the cylinder is (circumference) and the area of its side (curved surface area). First, we use these to find the height of the cylinder, and then we use the circumference to find its radius and then its diameter.

🎯 Exam Tip: When you have two equations with common variables (like \( 2\pi r \) in this case), dividing one equation by the other can often eliminate variables and simplify finding an unknown.

 

Question 2. A mansion has 12 right cylindrical pillars each having radius 50 cm and height 3.5 m. Find the cost of painting the lateral surface of the pillars at Rs. 20 per square metre.
Answer: Given, the radius of each cylindrical pillar (r) = 50 cm. We convert this to meters: \( r = 0.5 \text{ m} \).
The height of each pillar (h) = 3.5 m.
The curved surface area of one pillar (lateral surface area) = \( 2\pi rh \text{ sq. units} \).
Substitute the values:
\( \text{CSA of one pillar} = 2 \times \frac{22}{7} \times 0.5 \times 3.5 \)
\( = 2 \times 22 \times 0.5 \times 0.5 \)
\( = 22 \text{ sq. m} \)
Since there are 12 pillars, the total curved surface area of 12 pillars = \( 12 \times 22 = 264 \text{ sq. m} \).
The cost of painting per square metre = Rs. 20.
Total cost of painting = Total curved surface area \( \times \) cost per sq. m.
Total cost of painting = \( 264 \times \text{Rs. } 20 = \text{Rs. } 5280 \).
In simple words: We need to paint the sides of 12 pillars. First, we find the curved surface area of one pillar by converting measurements to meters. Then, we multiply that area by 12 for all pillars. Finally, we multiply this total area by the painting cost per square meter to get the total money needed.

🎯 Exam Tip: Always ensure all units are consistent (e.g., all in meters or all in centimeters) before performing calculations, especially in problems involving cost per unit area.

 

Question 3. The total surface area of a solid right circular cylinder is 231 cm². Its curved surface area is two thirds of the total surface area. Find the radius and height of the cylinder.
Answer: Given, the total surface area of the cylinder (TSA) = 231 cm².
The curved surface area (CSA) is two-thirds of the total surface area.
\( \text{CSA} = \frac{2}{3} \times \text{TSA} = \frac{2}{3} \times 231 = 2 \times 77 = 154 \text{ cm}^2 \).
We know the formula for CSA: \( 2\pi rh = 154 \text{ cm}^2 \) ....... (1)
We also know the formula for TSA: \( 2\pi r(h + r) = 231 \text{ cm}^2 \).
This can be written as \( 2\pi rh + 2\pi r^2 = 231 \).
Substitute the value of \( 2\pi rh \) from (1):
\( 154 + 2\pi r^2 = 231 \)
\( \implies 2\pi r^2 = 231 - 154 = 77 \)
\( \implies r^2 = \frac{77}{2\pi} = \frac{77}{2 \times \frac{22}{7}} = \frac{77 \times 7}{44} = \frac{7 \times 7}{4} = \frac{49}{4} \)
\( \implies r = \sqrt{\frac{49}{4}} = \frac{7}{2} = 3.5 \text{ cm} \)
Now, substitute the value of \( r \) back into equation (1) to find \( h \):
\( 2\pi rh = 154 \)
\( 2 \times \frac{22}{7} \times 3.5 \times h = 154 \)
\( 2 \times \frac{22}{7} \times \frac{7}{2} \times h = 154 \)
\( \implies 22 \times h = 154 \)
\( \implies h = \frac{154}{22} = 7 \text{ cm} \)
Therefore, the radius of the cylinder is 3.5 cm and its height is 7 cm.
In simple words: We start with the total surface area and the information that the curved area is two-thirds of it. First, we find the exact curved surface area. Then, using the total and curved surface area formulas, we find the radius. Once we have the radius, we use the curved surface area formula again to find the height. This helps us find the dimensions of the cylinder.

🎯 Exam Tip: Understand the relationship between total surface area (\( \text{TSA} = 2\pi r(h+r) \)) and curved surface area (\( \text{CSA} = 2\pi rh \)). Note that \( \text{TSA} = \text{CSA} + 2\pi r^2 \), which is often useful for solving problems like this.

 

Question 4. The total surface area of a solid right circular cylinder is 1540 cm². If the height is four times the radius of the base, then find the height of the cylinder.
Answer: Let the radius of the cylinder be \( r \).
Given condition: The height \( h \) is four times the radius, so \( h = 4r \).
The total surface area (TSA) of the cylinder is 1540 cm².
The formula for TSA of a cylinder is \( 2\pi r(h + r) = 1540 \).
Substitute \( h = 4r \) into the formula:
\( 2\pi r(4r + r) = 1540 \)
\( 2\pi r(5r) = 1540 \)
\( 10\pi r^2 = 1540 \)
\( 10 \times \frac{22}{7} \times r^2 = 1540 \)
\( \implies r^2 = \frac{1540 \times 7}{10 \times 22} = \frac{154 \times 7}{22} = 7 \times 7 = 49 \)
\( \implies r = \sqrt{49} = 7 \text{ cm} \)
Now, find the height using \( h = 4r \):
\( h = 4 \times 7 = 28 \text{ cm} \)
Therefore, the height of the cylinder is 28 cm.
In simple words: We are given the total surface area and a relationship between the cylinder's height and radius (height is four times the radius). We use the surface area formula, substitute the height in terms of radius, and then solve for the radius. Finally, we use the radius to find the height. This way, we determine both dimensions from the given information.

🎯 Exam Tip: When a relationship between dimensions is given (like \( h=4r \)), always substitute it into the main formula early on. This reduces the number of variables and simplifies the equation for solving.

 

Question 5. If the vertical angle of a right circular cone is 60° and its radius is 15 cm, then find its height and slant height.
Answer:
Given, in the cone OAB, OC is the height and AC is the radius. The line OC is perpendicular to AB.
The vertical angle of the cone is \( \angle AOB = 60^\circ \).
The semi-vertical angle is \( \angle AOC = \frac{60^\circ}{2} = 30^\circ \).
The radius \( AC = 15 \text{ cm} \).
In the right-angled triangle \( \triangle OAC \):
To find height (OC), use \( \tan(\angle AOC) = \frac{AC}{OC} \):
\( \tan 30^\circ = \frac{15}{OC} \)
\( \frac{1}{\sqrt{3}} = \frac{15}{OC} \)
\( \implies OC = 15\sqrt{3} \text{ cm} \)
So, the height of the cone (h) = \( 15\sqrt{3} \text{ cm} \).
To find slant height (OA), use \( \sin(\angle AOC) = \frac{AC}{OA} \):
\( \sin 30^\circ = \frac{15}{OA} \)
\( \frac{1}{2} = \frac{15}{OA} \)
\( \implies OA = 15 \times 2 = 30 \text{ cm} \)
Therefore, the slant height of the cone (l) = 30 cm.
In simple words: We draw a cone and split the top angle in half to make a right triangle. Knowing the radius and this half-angle, we use trigonometry (tangent to find height and sine to find slant height). This helps us find the cone's dimensions from its angle and base size.

🎯 Exam Tip: When a vertical angle is given for a cone, remember to use half of it (the semi-vertical angle) for trigonometric calculations in the right-angled triangle formed by the radius, height, and slant height.

 

Question 6. The central angle and radius of a sector of a circular disc are 180° and 21 cm respectively. If the edges of the sector are joined together to make a hollow cone, then find the radius of the cone.
Answer: