Volume & Surface Area

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The volume of a cuboid is length × width × height, and the volume of any prism is the area of its cross-section × its length. A cylinder is a circular prism, so its volume is πr²h. Surface area is the total area of all the faces: for a cuboid, add the three pairs of rectangles; for a cylinder, add the curved surface, 2πrh, to the two circular ends, 2πr².

Volume and surface area

Cuboid volume: l × w × h
Prism volume: area of cross-section × length
Cylinder volume: πr²h
Cuboid surface area: 2(lw + lh + wh)
Cylinder: curved surface 2πrh; total surface 2πr² + 2πrh

Cuboids

The volume of a cuboid is length × width × height, in cubic units such as cm³. The surface area is the total area of its six faces, which come in three matching pairs. A cuboid 7 cm by 4 cm by 5 cm has volume 140 cm³ and surface area 2(28 + 35 + 20) = 166 cm².

Prisms

A prism has the same cross-section all the way along. Its volume is the area of the cross-section multiplied by the length. For a triangular prism with a triangle of base 6 cm and height 5 cm, and length 12 cm, the cross-section is 15 cm² and the volume is 180 cm³.

Cylinders

A cylinder is a prism with a circular cross-section, so V = πr²h. Its curved surface unrolls into a rectangle of width 2πr and height h, so the curved surface area is 2πrh. The total surface area adds the two circles: 2πr² + 2πrh.

Finding a missing length

Rearrange the volume formula. A cylinder of volume 600 cm³ and radius 6 cm has height 600 ÷ (π × 36) = 5.31 cm. A prism of volume 180 cm³ and cross-section 15 cm² has length 12 cm.

Units and context

Volume uses cubic units and surface area uses square units. In context, 1 litre = 1000 cm³. Problems often ask how many small boxes fit in a larger one, or how much paint covers a surface; work out each quantity carefully and check the units match.

Worked examples

Example 1. A cuboid measures 7 cm by 4 cm by 5 cm. Work out its volume and surface area.

Volume: 7 × 4 × 5 = 140.
Faces: 7 × 4 = 28, 7 × 5 = 35, 4 × 5 = 20. Surface area: 2(28 + 35 + 20).

Answer: Volume 140 cm³; surface area 166 cm².

Example 2. A triangular prism has a cross-section with base 6 cm and height 5 cm, and length 12 cm. Work out its volume.

Cross-section: ½ × 6 × 5 = 15 cm².
15 × 12.

Answer: 180 cm³.

Example 3. A cylinder has radius 4 cm and height 9 cm. Work out its volume and curved surface area, to 1 decimal place.

V = π × 4² × 9 = 144π = 452.38…
Curved surface = 2π × 4 × 9 = 72π = 226.19…

Answer: Volume 452.4 cm³; curved surface area 226.2 cm².

Example 4. Work out the total surface area of the cylinder in the previous example, to 1 decimal place.

Two circles: 2 × π × 16 = 32π.
Total: 32π + 72π = 104π = 326.72…

Answer: 326.7 cm².

Example 5. A cylinder has volume 600 cm³ and radius 6 cm. Work out its height, to 2 decimal places.

600 = π × 36 × h.
h = 600 ÷ 36π = 5.305…

Answer: 5.31 cm.

Common mistakes

Using the diameter in πr²h. Use the radius.
Forgetting faces in surface area. A cuboid has six faces in three pairs; a cylinder has two ends as well as its curved surface.
Multiplying by the wrong length for a prism. Use the length perpendicular to the cross-section.
Mixing up units. Volume is cm³; surface area is cm².
Missing the ½ for a triangular cross-section. The cross-section area is ½ × base × height.

How volume and surface area are examined

Volume and surface area questions are common at both tiers in the 1MA1 and 4MA1 specifications; Higher papers add cones, spheres, pyramids and frustums. Questions ask you to: