The Volume of a Pyramid is One-Third that of a Prism

Ever wonder why the volume of a pyramid is a one-third the volume of a prism with the same base and height? Here is a way to show that it is true. To show that this cube has three times the volume of the pyramid, you can create three new pyramids that have the same volume as the original one. Assemble these three pyramids to form a cube.

Steps

1. Cut out three copies of the first pattern piece and one of the other as marked.
   
2. Fold and glue them into pyramids.
   
3. Stack the three identical pyramids to form a cube. Notice that all the pyramids have congruent bases and the same height, so they all have the same volume. This shows that a cube with the same base and height has three times the volume of the pyramid.
   

Notes

• This demonstration relies on the fact that two pyramids with congruent bases and equal heights have the same volume. To convince yourself that this is true, think about cutting a pyramid out of a stack of paper. You can shift the layers of paper into different orientations. You can make your pyramid slant in all sorts of directions. Of course, as you shift the paper, you don't change its volume so all pyramids you make have the same volume.