Surface Areas and Volumes

This chapter covers the measurement of 3D solids: their surface areas (total and lateral) and volumes.

Cuboid:
Lateral surface area = 2h(l + b)
Total surface area = 2(lb + bh + hl)
Volume = l x b x h

Cube (side a):
Lateral surface area = 4a²
Total surface area = 6a²
Volume = a³

Right Circular Cylinder (radius r, height h):
Curved surface area = 2 · pi · r · h
Total surface area = 2 · pi · r(r + h)
Volume = pi · r² · h

Key formulas

Right Circular Cone (radius r, height h, slant height l = root(r² + h²)):

Curved surface area = pi · r · l

Total surface area = pi · r(r + l)

Volume = (1/3) · pi · r² · h

Sphere (radius r):
Surface area = 4 · pi · r²
Volume = (4/3) · pi · r³

Hemisphere (radius r):
Curved surface area = 2 · pi · r²
Total surface area = 3 · pi · r²
Volume = (2/3) · pi · r³

Example 1

Find the volume of a cuboid with l = 5, b = 4, h = 3 cm.
Volume = 5 x 4 x 3 = 60 cubic cm.

Example 2

Total surface area of a cube of side 7 cm.
TSA = 6 x 7² = 6 x 49 = 294 sq cm.

Example 3

A cylinder has radius 7 cm and height 10 cm. Find curved surface area. (Use pi = 22/7.)
CSA = 2 x (22/7) x 7 x 10 = 440 sq cm.

Example 4

A cone has radius 6 cm and height 8 cm. Find slant height and curved surface area.
l = root(6² + 8²) = root(100) = 10 cm.
CSA = pi x 6 x 10 = 60 · pi approximately 188.57 sq cm.

Example 5

Volume of a sphere of radius 3 cm. (Use pi = 22/7.)
Volume = (4/3) x (22/7) x 27 = (4 x 22 x 27)/(3 x 7) = 2376/21 = 113.14 cubic cm approximately.

Example 6

Find the volume of a hemisphere of radius 10.5 cm. (pi = 22/7.)
Volume = (2/3) x (22/7) x (10.5)³ = (2/3) x (22/7) x 1157.625 = 2425.5 cubic cm approximately.

Example 7

A cylindrical tank has radius 2.1 m and height 4 m. How many litres does it hold? (1 cubic m = 1000 litres, pi = 22/7.)
Volume = (22/7) x 2.1² x 4 = (22/7) x 4.41 x 4 = 55.44 cubic m = 55440 litres.

Common mistakes

Students often use diameter instead of radius in formulas. Remember: radius = diameter/2. Also, for cones, the slant height l is NOT the same as the height h — always compute l = root(r² + h²) first.

Summary

Each 3D shape has formulas for lateral/curved surface area, total surface area and volume. Cone volume = (1/3) x cylinder volume with same dimensions. Sphere surface area and volume depend only on the radius. Always convert units before computing.

Practice Problems