Volume and surface area

Volume is the space inside a 3D shape, in cubic units such as cm³.

• Cuboid: volume = length × width × height (learn this)
• A prism has the same cross-section all the way through. Volume = area of cross-section × length (on the formulae sheet).

Example: find the volume of a cuboid 5 cm by 4 cm by 3 cm.

1. 5 × 4 × 3 = 60 cm³

Example: a prism has a triangular cross-section with base 6 cm and height 4 cm. It is 10 cm long. Find its volume.

1. Cross-section: ½ × 6 × 4 = 12 cm²
2. Volume = 12 × 10 = 120 cm³

A cylinder is a prism with a circle as its cross-section:

Volume of a cylinder = πr²h

Example: a cylinder has a radius of 3 cm and a height of 10 cm. Find its volume, to 1 decimal place.

1. Cross-section: π × 3² = 9π cm²
2. Volume = 9π × 10 = 90π
3. 282.7 cm³

Surface area is the total area of all the faces. Think of the net and add up the area of each part.

• Cuboid: 2 × (lw + lh + wh)
• Cylinder: two circles + a curved rectangle = 2πr² + 2πrh

Example: find the surface area of a cuboid 5 cm by 4 cm by 3 cm.

1. Pairs of faces: 5 × 4 = 20, 5 × 3 = 15 and 4 × 3 = 12
2. 2 × (20 + 15 + 12) = 94 cm²

Example: find the surface area of a cylinder with radius 3 cm and height 10 cm, to 1 decimal place.

1. Two circles: 2 × π × 3² = 18π
2. Curved surface: 2 × π × 3 × 10 = 60π
3. 18π + 60π = 78π = 245.0 cm²

These formulae are given on the formulae sheet or in the question:

• Sphere: volume = 4/3 πr³ and surface area = 4πr²
• Pyramid: volume = ⅓ × area of base × height

Example: a sphere has a radius of 6 cm. Find its volume and surface area, to 1 decimal place.

1. Volume = 4/3 × π × 6³ = 4/3 × π × 216 = 288π = 904.8 cm³
2. Surface area = 4 × π × 6² = 144π = 452.4 cm²

Example: a pyramid has a square base 6 cm by 6 cm and a height of 10 cm. Find its volume.

1. Base area: 6 × 6 = 36 cm²
2. ⅓ × 36 × 10 = 120 cm³

These formulae are given on the formulae sheet or in the question:

• Volume = ⅓πr²h
• Curved surface area = πrl, where l is the slant height (along the sloping side)

Example: a cone has a radius of 5 cm and a height of 12 cm. Find its volume and total surface area, to 1 decimal place.

1. Volume = ⅓ × π × 5² × 12 = 100π = 314.2 cm³
2. Slant height: l² = 5² + 12² = 169, so l = 13 cm (Pythagoras)
3. Curved surface = π × 5 × 13 = 65π
4. Base = π × 5² = 25π
5. Total = 90π = 282.7 cm²

A frustum is a cone with its top cut off parallel to the base. Its volume is the big cone's volume minus the small cone's.

1. Work out the volume of the whole cone.
2. Work out the volume of the small cone that was removed (it is similar to the big one).
3. Subtract.

Example: a cone has a radius of 6 cm and a height of 12 cm. The top 6 cm is cut off, leaving a frustum. The small cone removed has a radius of 3 cm. Find the volume of the frustum, to 1 decimal place.

1. Big cone: ⅓ × π × 6² × 12 = 144π
2. Small cone: ⅓ × π × 3² × 6 = 18π
3. 144π − 18π = 126π = 395.8 cm³