Volume of Pyramids, Cones and Spheres

Pyramids and cones: one third

A pyramid rises from a flat base to a single point. Fill it and pour it into the prism with the same base and height, and it fills exactly one third — so its volume is one third of that prism's:

V = \tfrac{1}{3} \times \text{base area} \times \text{height}

A cone is just a pyramid whose base is a circle. The base area of a circle is \pi r^2, so the same "one third" rule gives:

V = \tfrac{1}{3}\pi r^2 h

Each is exactly a third of the prism or cylinder built on the same base and height.

The sphere

A sphere is the perfectly round ball — every point on its surface the same distance r from the centre. It has its own volume formula, and a separate one for its surface area:

V = \tfrac{4}{3}\pi r^3 \qquad A = 4\pi r^2

pyramid V = \tfrac{1}{3} \times \text{base area} \times h;
cone V = \tfrac{1}{3}\pi r^2 h;
sphere volume V = \tfrac{4}{3}\pi r^3;
sphere surface area A = 4\pi r^2;
the constant \pi \approx 3.142.

See the three solids

Step through the figure: a cone, then a square-based pyramid, then a sphere. Each is labelled with its radius r (and the cone and pyramid with their height h).