Pyramid

Any cube can be divided into 6 pyramids. The next 3 pictures show this.

Once you understand this, we can prove the volume of a pyramid

First, we'll take the volume of the whole cube.

Volume of a Cube = Length x Width x Height

V_cube = 2h x 2h x 2h

or we could say that the base (B) of the pyramid is 2h x 2h, and we get

V_cube = (2h x 2h) x 2h

V_cube = 2Bh

Now, since there are 6 pyramids in a cube, we can divide by 6 to find the volume of 1 pyramid.

V_pryamid = (1/6)V_cube

V_pryamid = (1/6)2Bh

V_pryamid = (1/3)Bh

Or if we want to write it fancy,

And that's one of many, many ways to show the volume of a pyramid.

What about if the pyramid is slanted? The formula still works, due to Cavelieri's Principle

As you can see, the slabs making up the pyramid are the same size, just shifted over, so the volume will be the same for both

What if the base of the pyramid isn't a square? Turns out it doesn't matter what shape the base of the pyramid is. As long as the bases and heights are the same, the areas of the cross sections will be equal, and thus the volume will still be (1/3)Bh