Surface Areas and Volumes Solved Examples (Class 10 Maths)
Surface area is the total area of all faces of a 3D shape, while volume is the amount of space it occupies. These concepts are essential for calculating ma
TL;DR: Surface area is the total area of all faces of a 3D shape, while volume is the amount of space it occupies. These concepts are essential for calculati…
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Surface area is the total area of all faces of a 3D shape, while volume is the amount of space it occupies. These concepts are essential for calculating ma
Surface Areas and Volumes — Solved Numerical Examples (Step by Step)
Example 1: Find the surface area of a cube with side length 4 cm.
Solution: Given:
Side length a = 4 cm
For a cube:
Surface area = 6a^2
Surface area = 6 × (4)^2
Surface area = 6 × 16
Surface area = 96 cm^2
Example 2: Calculate the volume of a rectangular prism with dimensions 5 cm, 6 cm, and 8 cm.
Solution: Given:
Length l = 5 cm
Breadth b = 6 cm
Height h = 8 cm
For a rectangular prism:
Volume = l × b × h
Volume = 5 × 6 × 8
Volume = 240 cm^3
Example 3: Find the total surface area of a cylinder with radius 3 cm and height 10 cm.
Solution: Given:
Radius r = 3 cm
Height h = 10 cm
For a cylinder:
Total surface area = 2πr(r + h)
Total surface area = 2 × 3.14 × 3 × (3 + 10)
Total surface area = 2 × 3.14 × 3 × 13
Total surface area = 6.28 × 39
Total surface area = 244.92 cm^2
Example 4: Find the volume of a sphere with radius 5 cm.
Solution: Given:
Radius r = 5 cm
For a sphere:
Volume = (4/3)πr^3
Volume = (4/3) × 3.14 × (5)^3
Volume = (4/3) × 3.14 × 125
Volume = 4.19 × 125
Volume = 523.75 cm^3
Example 5: Calculate the volume of a cone with radius 4 cm and height 9 cm.
Solution: Given:
Radius r = 4 cm
Height h = 9 cm
For a cone:
Volume = (1/3)πr^2h
Volume = (1/3) × 3.14 × (4)^2 × 9
Volume = (1/3) × 3.14 × 16 × 9
Volume = (1/3) × 452.16
Volume = 150.72 cm^3
Example 6: Find the curved surface area of a cone with radius 6 cm and slant height 10 cm.
Solution: Given:
Radius r = 6 cm
Slant height l = 10 cm
For a cone:
Curved surface area = πrl
Curved surface area = 3.14 × 6 × 10
Curved surface area = 3.14 × 60
Curved surface area = 188.4 cm^2
Example 7: A rectangular tank is 8 m long, 5 m wide, and 4 m deep. If it is filled with water up to a height of 3 m, find the volume of water in the tank.
Solution: Given:
Length l = 8 m
Width w = 5 m
Height of water h = 3 m (not 4 m, as tank is only filled to 3 m)
Volume of water = l × w × h
Volume of water = 8 × 5 × 3
Volume of water = 120 m^3
Tips
- Always distinguish between total surface area (all faces including top and bottom) and curved surface area (only the curved part). For cylinders and cones, remember they have a flat circular base.
- Use the correct formulas for slant height and vertical height in cones. Slant height is used for curved surface area, while vertical height is used for volume.
- For composite shapes, calculate the surface area and volume of each individual part and then add or subtract appropriately.
Frequently Asked Questions
What is the difference between total surface area and curved surface area of a cylinder?
Curved surface area of a cylinder is 2πrh (just the side). Total surface area is 2πrh + 2πr² (side plus both circular bases). When asked for total surface area, include the top and bottom circles.
How do I find the slant height of a cone if only radius and height are given?
Use the Pythagoras theorem: slant height l = sqrt(r² + h²), where r is the radius and h is the vertical height. The slant height, radius, and height form a right triangle.
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