# Splitting a cube into pyramids

- Author:
- Brad Ballinger

Suppose the cube has side length 12. What is the volume of the blue pyramid, and how do you know?

## How I want to improve this experience

0. Fix the question so that it acknowledges correct answers.
1. Use equal base & height arguments to show the same result* holds for a triangular pyramid inscribed in a triangular prism (it's not as smooth as the cube case).
2. Use stretches and shears to show that the same result* holds for any pyramid inscribed in a parallelepiped (right or oblique prism over a parallelogram base).
3. By triangulating the base and applying #1, show that the same result* holds for any pyramid inscribed in a prism, regardless of how many edges the base has.
4. Use Cavalieri's Principle to show that the same result* holds for any cone inscribed in a cylinder.
5. Address double cones.
* The result I'm talking about is that a pyramid (more generally, cone) has one third the volume of a prism (more generally, cylinder) with the same base and height.