Maharashtra Board Class 9 Maths Part 2 Geometry Chapter 9 Set 9.1 Surface Area and Volume Solutions

Question 1. Length, breadth and height of a cuboid shape box of medicine is 20 cm, 12 cm and 10 cm respectively. Find the surface area of vertical faces and total surface area of this box. Given: For cuboid shape box of medicine, length \( (l) = 20 \) cm, breadth \( (b) = 12 \) cm and height \( (h) = 10 \) cm. To find: Surface area of vertical faces and total surface area of the box
Answer: Solution: i. Surface area of vertical faces of the box \( = 2(l + b) \times h \) \( = 2(20 + 12) \times 10 \) \( = 2 \times 32 \times 10 \) \( = 640 \) sq.cm.
ii. Total surface area of the box \( = 2 (lb + bh + lh) \) \( = 2(20 \times 12 + 12 \times 10 + 20 \times 10) \) \( = 2(240 + 120 + 200) \) \( = 2 \times 560 \) \( = 1120 \) sq.cm.
\( \therefore \) The surface area of vertical faces and total surface area of the box are 640 sq.cm, and 1120 sq.cm, respectively.
In simple words: To find the surface area of vertical faces, we use the formula \(2(l+b)h\), and for total surface area, we use \(2(lb+bh+lh)\). Substituting the given dimensions, we calculate these values for the cuboid.

🎯 Exam Tip: Remember the specific formulas for vertical and total surface area of a cuboid. Accurate substitution and calculation are crucial for scoring full marks.

Question 2. Total surface area of a box of cuboid shape is 500 sq.unit. Its breadth and height is 6 unit and 5 unit respectively. What is the length of that box? Given: For cuboid shape box, breadth \( (b) = 6 \) unit, height \( (h) = 5 \) unit Total surface area \( = 500 \) sq. unit. To find: Length of the box \( (l) \)
Answer: Solution: Total surface area of the box \( = 2 (lb + bh + lh) \)
\( \therefore 500 = 2 (6l + 6 \times 5 + 5l) \)
\( \therefore \frac{500}{2} = (11l + 30) \)
\( \therefore 250 = 11l + 30 \)
\( \therefore 250 - 30 = 11l \)
\( \therefore 220 = 11l \)
\( \therefore 220 = l \)
\( \therefore \frac{220}{11} = l \)
\( \therefore l = 20 \) units
\( \therefore \) The length of the box is 20 units.
In simple words: Given the total surface area, breadth, and height of a cuboid, we use the total surface area formula to set up an equation. By substituting the known values and solving for length, we find the missing dimension.

🎯 Exam Tip: When a dimension is unknown, substitute the given values into the appropriate formula and solve the resulting algebraic equation systematically. Clearly show each step of your calculation.

Question 3. Side of a cube is 4.5 cm. Find the surface area of all vertical faces and total surface area of the cube. Given: Side of cube \( (l) = 4.5 \) cm To find: Surface area of all vertical faces and the total surface area of the cube
Answer: Solution: i. Area of vertical faces of cube \( = 4l^2 \) \( = 4 (4.5)^2 \) \( = 4 \times 20.25 \) \( = 81 \) sq.cm.
ii. Total surface area of the cube \( = 6l^2 \) \( = 6 (4.5)^2 \) \( = 6 \times 20.25 \) \( = 121.5 \) sq.cm.
\( \therefore \) The surface area of all vertical faces and the total surface area of the cube are 81 sq.cm, and 121.5 sq.cm, respectively.
In simple words: For a cube, the area of vertical faces is \(4l^2\) and the total surface area is \(6l^2\). We substitute the given side length to calculate both areas.

🎯 Exam Tip: Distinguish between the formula for vertical surface area (lateral surface area) and total surface area for a cube. Pay attention to the squaring of the side length.

Question 4. Total surface area of a cube is 5400 sq. cm. Find the surface area of all vertical faces of the cube. Given: Total surface area of cube \( = 5400 \) sq.cm. To find: Surface area of all vertical faces of the cube
Answer: Solution: i. Total surface area of cube \( = 6l^2 \)
\( \therefore 5400 = 6l^2 \)
\( \therefore \frac{5400}{6} = l^2 \)
\( \therefore l^2 = 900 \)
ii. Area of vertical faces of cube \( = 4l^2 \) \( = 4 \times 900 \) \( = 3600 \) sq.cm.
\( \therefore \) The surface area of all vertical faces of the cube is 3600 sq.cm.
In simple words: We first find \(l^2\) using the given total surface area of the cube. Then, we use this value of \(l^2\) to calculate the surface area of all vertical faces, which is \(4l^2\).

🎯 Exam Tip: Understand that if the total surface area is given, you can first find \(l^2\) and then use it to find the area of vertical faces without calculating \(l\) itself. This saves time and avoids potential square root errors.

Question 5. Volume of a cuboid is 34.50 cubic metre. Breadth and height of the cuboid is 1.5 m and 1.15 m respectively. Find its length. Given: Breadth \( (b) = 1.5 \) m, height \( (h) = 1.15 \) m Volume of cuboid \( = 34.50 \) cubic metre To find: Length of the cuboid \( (l) \)
Answer: Solution: Volume of cuboid \( = l \times b \times h \)
\( \therefore 34.50 = l \times b \times h \)
\( \therefore 34.50 = l \times 1.5 \times 1.15 \)
\( \therefore l = \frac{34.50}{1.5 \times 1.15} \)
\( \therefore l = \frac{34500}{15 \times 115} \)
\( \therefore l = \frac{300}{15} \)
\( \therefore l = 20 \)
\( \therefore \) The length of the cuboid is 20 m.
In simple words: Given the volume, breadth, and height of a cuboid, we use the formula for volume \( (V = l \times b \times h) \) to solve for the unknown length.

🎯 Exam Tip: Be careful with decimal calculations. Converting decimals to fractions or multiplying by powers of 10 to clear decimals can simplify the division process. Always include units in your final answer.

Question 6. What will be the volume of a cube having length of edge 7.5 cm ? Given: Length of edge of cube \( (l) = 7.5 \) cm To find: Volume of a cube
Answer: Solution: Volume of a cube \( = l^3 \) \( = (7.5)^3 \) \( = 421.875 \approx 421.88 \) cubic cm
\( \therefore \) The volume of the cube is 421.88 cubic cm.
In simple words: The volume of a cube is found by cubing its side length. We calculate \(7.5^3\) to get the volume.

🎯 Exam Tip: Ensure you know the difference between surface area formulas and volume formulas for cubes. Practice cubing decimal numbers carefully, and remember to round off appropriately if required.

Question 7. Radius of base of a cylinder is 20 cm and its height is 13 cm, find its curved surface area and total surface area, \( (\pi = 3.14) \) Given: Radius \( (r) = 20 \) cm, height \( (h) = 13 \) cm To find: Curved surface area and the total surface area of the cylinder
Answer: Solution: i. Curved surface area of cylinder \( = 2\pi rh \) \( = 2 \times 3.14 \times 20 \times 13 \) \( = 1632.8 \) sq.cm
ii. Total surface area of cylinder \( = 2\pi r(r + h) \) \( = 2 \times 3.14 \times 20(20 + 13) \) \( = 2 \times 3.14 \times 20 \times 33 \) \( = 4144.8 \) sq.cm
\( \therefore \) The curved surface area and the total surface area of the cylinder are 1632.8 sq.cm and 4144.8 sq.cm respectively.
In simple words: Using the given radius and height, we apply the formulas \(2\pi rh\) for curved surface area and \(2\pi r(r+h)\) for total surface area of the cylinder, substituting \( \pi = 3.14 \).

🎯 Exam Tip: Clearly differentiate between the formulas for curved surface area and total surface area of a cylinder. Use the specified value of \( \pi \) and perform multiplications accurately.

Question 8. Curved surface area of a cylinder is 1980 cm\(^2\) and radius of its base is 15 cm. Find the height of the cylinder. \( (\pi = \frac{22}{7}) \) Given: Curved surface area of cylinder \( = 1980 \) sq.cm., radius \( (r) = 15 \) cm To find: Height of the cylinder \( (h) \)
Answer: Solution: Curved surface area of cylinder \( = 2\pi rh \)
\( \therefore 1980 = 2 \times \frac{22}{7} \times 15 \times h \)
\( \therefore h = \frac{1980 \times 7}{2 \times 22 \times 15} \)
\( \therefore h = 21 \) cm
\( \therefore \) The height of the cylinder is 21 cm.
In simple words: We use the formula for the curved surface area of a cylinder, \(2\pi rh\). By substituting the given curved surface area, radius, and \( \pi \), we can solve for the unknown height \(h\).

🎯 Exam Tip: When a specific value for \( \pi \) is provided (like \( \frac{22}{7} \)), ensure you use that value. Rearranging the formula correctly to solve for the unknown variable is a key skill here.