Pyramids, cones and spheres

The pyramid

A pyramid is a three-dimensional shape whose base is a polygon (triangle, square, pentagon, etc.) and whose lateral faces are triangles that meet at a single point, the apex. The height is the distance between the apex and the plane of the base, measured perpendicularly.

Volume of a pyramid: V = (1/3) × Base_area × Height

Example: a pyramid with a square base of side length 6 cm and a height of 9 cm.

Base_area = 6² = 36 cm²

V = (1/3) × 36 × 9 = 108 cm³

The cone of revolution

A cone of revolution is formed by rotating a right-angled triangle about one of the two sides of the right angle. Its base is a disc of radius r, and its height h connects the apex to the centre of the base.

Volume: V = (1/3) × π × r² × h

The sphere

A sphere of radius r is the set of all points in space situated at a distance r from a centre O. The ball is the solid figure bounded by the sphere.

• Volume of the sphere: V = (4/3) × π × r³
• Surface area of the sphere: A = 4 × π × r²

Numerical example

Sphere with radius r = 5 cm:

V = (4/3) × π × 5³ = (4/3) × π × 125 ≈ 523.6 cm³

Solid	Volume	Surface area
Pyramid	(1/3) × Base_area × h	depends on the faces
Cone	(1/3) × π × r² × h	π × r × (r + slant height)
Sphere	(4/3) × π × r³	4 × π × r²

Common pitfall

The factor 1/3 is often overlooked for pyramids and cones: we multiply the area of the base by the height as we would for a prism, which gives a result that is three times too large. Always remember to check whether the solid is a prism/cylinder (no 1/3) or a pyramid/cone (factor of 1/3).