## IM Commentary

The goal of this task is to calculate the volume of a particular pyramid with square base, which is easily reasoned by viewing it as one sixth of a cube. The method does not apply to other pyramids or to cones but has the advantage of being very concrete. Students might build different shaped pyramids and see what happens when they try to put six of them together. Only when the height of the pyramid is half the length of the sides of the base will they fit together as in this task. It is remarkable that the formula for the volume obtained here, $\frac{1}{3}{\rm area(base)} \times {\rm height}$, applies to all pyramids with square base.

Understanding the formula for the volume of a pyramid is the first step toward the analogous forumla for a cylinder. Although it is not possible to fit cones together to make a cylinder, the ratio of the volume of a cone to the volume of the cylinder (with equal base and the same height) is the same as the ratio of the square pyramid to the cube (or rectangular prism it is inscribed in). We can see this by taking horizontal slices: for the cone this is circles while for the square pyramid it is squares but in both cases the ratio (area of slice: area of slice of circumscribed shape) is the same for every slice. If we view the three dimensional objects as made up of many very thin slices in this way, this helps explain why the volume of a cone is $\frac{1}{3}$ of the cylinder in which it is inscribed. This idea is called Cavalieri's principle and the basic idea dates back at least to Archimedes.