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). To explore how shear transformations affect area in 2D, see this game about manipulating a triangle by shear transformations. A similar thing happens to volumes (but no longer areas) under shear transformations in 3D. Stretches in 2D can distort lengths inconsistently, but they multiply all areas by the same factor, thereby preserving ratios between them. Likewise, stretches in 3D can distort areas inconsistently, but they multiply all volumes by the same factor, preserving ratios between them. 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.