Splitting a cube into pyramids
- 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.