What is a platonic solid and how many are there and what are their names?

Answer 1

A Platonic solid is the 3-D shape equivalent of a polygon: it is a three dimensional figure whose sides are congruent, regular polygons, with identical vertices. Unlike the 2-dimensional case (in which there are infinitely many polygons) there are only 5 Platonic solids: * The tetrahedron, which has 4 triangular sides. * The cube (or hexahedron), which has 6 square sides. * The octahedron, which has 8 triangular sides. * The dodecahedron, which has 12 pentagonal sides. * The icosahedron, which has 20 triangular sides. Here is how the 5 Platonic solids were found, and how we know there aren't any more: Think about the sum of the angles at a vertex (by the definition of a Platonic solid, all vertices are identical). In the plane, angles around a vertex add up to 360 degrees, but we don't want the vertex to lie flat - otherwise, we'd end up with a huge flat sheet instead of a polyhedron. We also want at least 3 polygons around a vertex, because otherwise the result will become a flat figure without volume. If the sides are triangles, we can have 3 triangles around a vertex (getting the tetrahedron), 4 triangles around a vertex (getting the octahedron), or 5 triangles around a vertex (getting the icosahedron). We can't have 6 or more, because then the sum of angles wouldn't be less than 360. If the sides are squares, we can have 3 squares around a vertex, getting the cube. 4 squares around a vertex would mean the sum of angles is 360, and 5 or more is even more impossible. Finally, we can take 3 pentagons around a vertex, getting the dodecahedron; more pentagons will give us an angle sum of over 360. We can't use any shapes with more than 6 sides, because their angles are larger and we can't even fit 3 around a vertex. Even 3 hexagons will give an angle sum of 360 degrees, and anything more than that is even worse.