12/12/2023 0 Comments Octahedron angles between facesAnd, obviously, no polygon with more than six sides can be used either, because the interior angles just keep getting larger. For this reason we can use hexagons to make a tesselation of the plane, but we cannot use them to make a Platonic solid. What about the regular hexagon, that is, the six-sided figure? Well, its interior angles are 120°, so if we fit three of them together at a vertex the angles sum to precisely 360°, and therefore they lie flat, just like four squares (or six equilateral triangles) would do. (We could fit four squares together, but then they would lie flat, giving us a tesselation instead of a solid.) The interior angles of the regular pentagon are 108°, so again we can fit only three together at a vertex, giving us the dodecahedron.Īnd that makes five regular polyhedra. Each interior angle of a square is 90°, so we can fit only three of them together at each vertex, giving us a cube. Now, each interior angle of an equilateral triangle is 60°, hence we could fit together three, four, or five of them at a vertex, and these correspond to to the tetrahedron, the octahedron, and the icosahedron. Second, observe that the sum of the interior angles of the faces meeting at each vertex must be less than 360°, for otherwise they would not all fit together. However, it is not difficult to show that there must be five-and that there cannot be more than five.įirst, consider that at each vertex (point) at least three faces must come together, for if only two came together they would collapse against one another and we would not get a solid. It is natural to wonder why there should be exactly five Platonic solids, and whether there might conceivably be one that simply hasn't been discovered yet. Twenty triangular faces, twelve vertices, and thirty edges. Twelve pentagonal faces, twenty vertices, and thirty edges. Six square faces, eight vertices, and twelve edges.Įight triangular faces, six vertices, and twelve edges. “Polyhedra” is a Greek word meaning “many faces.” There are five of these, and they are characterized by the fact that each face is a regular polygon, that is, a straight-sided figure with equal sides and equal angles:įour triangular faces, four vertices, and six edges. The so-called Platonic Solids are convex regular polyhedra.
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