Cones, spheres, pyramids and composite solids

Cones, spheres and pyramids have their own formulae (usually given). Composite solids are made by joining simple ones together.

GCSE · 1MA1 · G16, G17
International GCSE · 4MA1 · 4.10
About 17 minutes

Practise this lesson free (6 questions)

The formulae

Cone: volume 13πr^2h, curved surface area πrl, where l is the slant height.
Sphere: volume 43πr³, surface area 4πr².
Pyramid: volume 13 × base area × height.
A hemisphere is half a sphere: volume 23πr³, curved surface 2πr².

Solid	Volume	Surface
Cone	13πr^2h	πrl + πr²
Sphere	43πr³	4πr²
Hemisphere	23πr³	2πr² + πr²
Pyramid	13 × base × h	add the faces

Slant height

The radius, the vertical height and the slant height of a cone make a right-angled triangle.
r = 5, h = 12: l = √(5² + 12²) = 13.
Curved surface = π × 5 × 13 = 65π = 204.2 cm².

Composite solids

Add the volumes of the parts. For surface area, include only the outside surfaces.
A cylinder (r = 3, h = 10) with a hemisphere on top:
Volume = π(9)(10) + 23π(27) = 90π + 18π = 108π = 339.3 cm³.

Worked example

A cone has radius 5 cm and height 12 cm. Find its curved surface area.

Slant height l = √(25 + 144) = 13
πrl = π × 5 × 13 = 65π

Answer: 204.2 cm² (or 65π)

Key idea

Use the slant height for a cone's curved surface. A hemisphere is half a sphere plus a flat circle. For composite solids, add volumes and add only the outside surfaces.

Extra practice

2 more questions

Try each one, then tap to check.

A cone has radius 24 cm and height 10 cm. Find its slant height.
Show the answer
Answer: 26 cm
√(24² + 10²) = 26 cm.
A pyramid has a square base of side 12 cm and a vertical height of 10 cm. Find its volume.
Show the answer
Answer: 480 cm³
13 × 144 × 10 = 480 cm³.

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Sources and references

Checked by the GCSE-ready editorial team, 17 October 2026. Last updated 2 October 2026. Spotted a mistake? Tell us and we will fix it.