Before discussing what types of polyhedrons there are, it’s necessary to define what makes a shape a polyhedron. A polyhedron is a solid with flat faces and no curves. A pyramid is a polyhedron, but a cylinder is not.
Whether a shape is a polyhedron can be confirmed with an exercise called the polyhedral formula, which was developed in the 1700s by Swiss mathematician Leonhard Euler. The formula states that if you add the number of faces and the number of corners and subtract the number of edges, the result will always be the number two. You can try it with a cube. A cube has six faces, eight corners, and twelve edges (6 + 8 - 12 = 2).
One type of polyhedron is the pyramid. Whatever shape the base has, the sides are triangles of varying shapes. The pyramids that are most familiar to people are the ancient pyramids of Egypt. These pyramids have square bases. But you could also build a pyramid with an octagon as its base. Thus you would have an octagon with eight triangles as the sides. You might see a skylight on top of a building in this shape. You can check the formula to confirm that this is still a polyhedron. Nine faces plus nine corners minus sixteen edges equals two (9 + 9 - 16 = 2).
Another type of polyhedron is a prism. A prism has the same shape all the way across the length of the object. For example, a triangular prism has a rectangle for the length of all three sides with a triangle at each end. If you cut the prism in half, the new pieces would still have a triangle at each end and squares or rectangles along the three sides. Check the formula once again to confirm that it’s a polyhedron. Five faces plus six corners minus nine edges equals two (5 + 6 - 9 = 2).
There is one more type of polyhedron called a platonic solid. Platonic refers to plates, meaning flat faces. There are five shapes in this category - tetrahedron, cube, octahedron, dodecahedron, and icosahedron. The tetrahedron is made of four identical triangles. A cube has six identical squares. An octahedron is two square-base pyramids with the bases joined. A dodecahedron has twelve identical pentagons. Finally, an icosahedron has twenty identical triangles. Even with this complex shape, the math in the formula still works. Twenty faces plus twelve corners minus thirty edges equals two (20 + 12 - 30 = 2).
The next time you see a photo of the pyramids you will remember that a pyramid is also a polyhedron and you’ll know the formula to test it to be sure.
