Simple proof of the volume of a pyramid

The full proof of the formula of the volume of a pyramid is beyond this course but here is a simple illustration to show that the volume of a pyramid is a third of the area of its base multiplied by its height.

Here is a cube with a point in its centre

We can create a pyramid within the cube by drawing lines from all corners of the base of the cube to the point, like this

We could create five other congruent pyramids by drawing lines from the corners of the other five faces to the centre of the cube. The volume of the cube would then be filled by six congruent pyramids. The volume of each pyramid would be one sixth the volume of the cube.

Let = the length of each side of the cube

Let = the area of each face of the cube = the area of the base of each pyramid

Volume of the cube =

Let = Volume of each pyramid

The length of each side of the cube is twice the vertical height of the pyramid

Height of each pyramid =

Let = the vertical height of each pyramid