Maharashtra Board Class 8 Maths Chapter 16 Surface Area and Volume Set 16.2 Solutions

Question 1. In each example given below, radius of base of a cylinder and its height are given. Then find the curved surface area and total surface area.
(i) r = 7 cm, h = 10 cm
(ii) r = 1.4 cm, h = 2.1 cm
(iii) r = 2.5 cm, h = 7 cm
(iv) r = 70 cm, h = 1.4 cm
(v) r = 4.2 cm, h = 14 cm
Answer:
i. Given: r = 7 cm and h = 10 cm
To find: Curved surface area of cylinder and total surface area
Curved surface area of the cylinder = \(2\pi rh\)
= \(2 \times \frac{22}{7} \times 7 \times 10\)
= \(2 \times 22 \times 10\)
= \(440\) sq.cm
Total surface area of the cylinder:
= \(2\pi r(h + r)\)
= \(2 \times \frac{22}{7} \times 7(10 + 7)\)
= \(2 \times \frac{22}{7} \times 7 \times 17\)
= \(2 \times 22 \times 17\)
= \(748\) sq.cm
The curved surface area of the cylinder is \(440\) sq.cm and its total surface area is \(748\) sq.cm.
ii. Given: r = 1.4 cm and h = 2.1 cm
To find: Curved surface area of cylinder and total surface area
Curved surface area of the cylinder = \(2\pi rh\)
= \(2 \times \frac{22}{7} \times 1.4 \times 2.1\)
= \(2 \times 22 \times 0.2 \times 2.1\)
= \(18.48\) sq.cm
Total surface area of the cylinder = \(2\pi r (h + r)\)
= \(2 \times \frac{22}{7} \times 1.4 (2.1 + 1.4)\)
= \(2 \times \frac{22}{7} \times 1.4 \times 3.5\)
= \(2 \times 22 \times 0.2 \times 3.5\)
= \(30.80\) sq.cm
.: The curved surface area of the cylinder is \(18.48\) sq.cm and its total surface area is \(30.80\) sq.cm.
iii. Given: r = 2.5 cm and h = 7 cm
To find: Curved surface area of cylinder and total surface area
Curved surface area of the cylinder = \(2\pi rh\)
= \(2 \times \frac{22}{7} \times 2.5 \times 7\)
= \(2 \times 22 \times 2.5\)
= \(110\) sq.cm
Total surface area of the cylinder = \(2\pi r(h + r)\)
= \(2 \times \frac{22}{7} \times 2.5 (7 + 2.5)\)
= \(2 \times \frac{22}{7} \times 2.5 \times 9.5\)
= \(\frac{1045}{7}\)
= \(149.29\) sq.cm
.: The curved surface area of the cylinder is \(110\) sq.cm and its total surface area is \(149.29\) sq.cm.
iv. Given: r = 70 cm and h = 1.4 cm
To find: Curved surface area of cylinder and total surface area
Curved surface area of the cylinder = \(2\pi rh\)
= \(2 \times \frac{22}{7} \times 70 \times 1.4\)
= \(2 \times 22 \times 10 \times 1.4\)
= \(616\) sq.cm
Total surface area of the cylinder = \(2\pi r(h + r)\)
= \(2 \times \frac{22}{7} \times 70(1.4 + 70)\)
= \(2 \times \frac{22}{7} \times 70 \times 71.4\)
= \(2 \times 22 \times 10 \times 71.4\)
= \(2 \times 22 \times 714\)
= \(31416\) sq.cm
.: The curved surface area of the cylinder is \(616\) sq.cm and its total surface area is \(31416\) sq.cm.
v. Given: r = 4.2 cm and h = 14 cm
To find: Curved surface area of cylinder and total surface area
Curved surface area of the cylinder = \(2\pi rh\)
= \(2 \times \frac{22}{7} \times 4.2 \times 14\)
= \(2 \times 22 \times 4.2 \times 2\)
= \(369.60\) sq.cm
Total surface area of the cylinder = \(2\pi r (h + r)\)
= \(2 \times \frac{22}{7} \times 4.2 (14 + 4.2)\)
= \(2 \times \frac{22}{7} \times 4.2 \times 18.2\)
= \(2 \times 22 \times 0.6 \times 18.2\)
= \(480.48\) sq.cm
.: The curved surface area of the cylinder is \(369.60\) sq.cm and its total surface area is \(480.48\) sq.cm.
In simple words: For each given cylinder, we applied the formulas for curved surface area (\(2\pi rh\)) and total surface area (\(2\pi r(h+r)\)) using the provided radius and height values to find the respective areas.

🎯 Exam Tip: Remember to clearly state the formulas used and substitute the values correctly. Pay attention to units (sq.cm) in your final answer and ensure calculations are precise, especially with decimals.

Question 2. Find the total surface area of a closed cylindrical drum if its diameter is 50 cm and height is 45 cm. (\(\pi = 3.14\))
Answer:
Given: For cylindrical drum:
Diameter (d) = 50 cm
and height (h) = 45 cm
To find: Total surface area of the cylindrical drum
Solution:
Diameter (d) = 50 cm

\( \implies \) radius (r) = \(\frac{d}{2} = \frac{50}{2} = 25\) cm
Total surface area of the cylindrical drum = \(2\pi r (h + r)\)
= \(2 \times 3.14 \times 25 (45 + 25)\)
= \(2 \times 3.14 \times 25 \times 70\)
= \(10,990\) sq.cm

\( \implies \) The total surface area of the cylindrical drum is \(10,990\) sq.cm.
In simple words: We first found the radius from the given diameter, then applied the formula for the total surface area of a closed cylinder, \(2\pi r(h+r)\), using the given height and calculated radius.

🎯 Exam Tip: Ensure you use the correct value for \(\pi\) as specified in the question (e.g., 3.14 or \(\frac{22}{7}\)). Double-check unit conversions and calculations for accuracy to avoid errors in the final result.

Question 3. Find the area of base and radius of a cylinder if its curved surface area is 660 sq.cm and height is 21 cm.
Answer:
Given: Curved surface area = 660 sq.cm, and height = 21 cm
To find: area of base and radius of a cylinder
Solution:
i. Curved surface area of cylinder = \(2\pi rh\)

\( \implies \) \(660 = 2 \times \frac{22}{7} \times r \times 21\)

\( \implies \) \(660 = 2 \times 22 \times r \times 3\)

\( \implies \) \(\frac{660}{2 \times 22 \times 3} = r\)

\( \implies \) \(\frac{660}{66} = r\)

\( \implies \) \(5 = r\)
i.e., r = 5 cm
ii. Area of a base of the cylinder = \(\pi r^2\)
= \(\frac{22}{7} \times 5 \times 5\)
= \(\frac{550}{7}\)
= \(78.57\) sq.cm

\( \implies \) The radius of the cylinder is 5 cm and the area of its base is \(78.57\) sq.cm.
In simple words: We used the given curved surface area and height with the formula \(2\pi rh\) to first find the radius, and then calculated the base area using the radius in the formula \(\pi r^2\).

🎯 Exam Tip: When given an area and one dimension, use the appropriate formula to work backward and find the missing dimension. Clearly show each step of your calculation, especially when solving for variables.

Question 4. Find the area of the sheet required to make a cylindrical container which is open at one side and whose diameter is 28 cm and height is 20 cm. Find the approximate area of the sheet required to make a lid of height 2 cm for this container.
Answer:
Given: For cylindrical container:
diameter (d) = 28 cm, height (\(h_{1}\)) = 20 cm
For cylindrical lid: height (\(h_{2}\)) = 2 cm
To find:
i. Surface area of the cylinder with one side open
ii. Area of sheet required to made a lid
Solution:
diameter (d) = 28 cm

\( \implies \) radius (r) = \(\frac{d}{2} = \frac{28}{2} = 14\) cm
i. Surface area of the cylinder with one side open = Curved surface area + Area of a base
= \(2\pi rh_{1} + \pi r^2\)
= \(\pi r (2h_{1} + r)\)
= \(\frac{22}{7} \times 14 \times (2 \times 20 + 14)\)
= \(22 \times 2 \times (40 + 14)\)
= \(22 \times 2 \times 54\)
= \(2376\) sq.cm
ii. Area of sheet required to made a lid = Curved surface area of lid + Area of upper surface
= \(2\pi rh_{2} + \pi r^2\)
= \(\pi r (2h_{2} + r)\)
= \(\frac{22}{7} \times 14 \times (2 \times 2 + 14)\)
= \(22 \times 2 \times (4 + 14)\)
= \(22 \times 2 \times 18\)
= \(792\) sq cm

\( \implies \) The area of the sheet required to make the cylindrical container is \(2376\) sq. cm and the approximate area of a sheet required to make the lid is \(792\) sq. cm.
In simple words: We calculated the radius from the diameter. Then, for the open container, we summed the curved surface area and one base area. For the lid, we added its curved surface area and one base area using the relevant heights.

🎯 Exam Tip: Carefully read whether a container is open or closed, as this affects the total surface area formula (an open container will lack one base area). Differentiate between the container's height and the lid's height for accurate calculations.