Regular icosahedron

In geometry, a regular icosahedron (ˌaɪkɒsəˈhiːdrən,-kə-,-koʊ- or aɪˌkɒsəˈhiːdrən) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. It is one of the five Platonic solids, and the one with the most faces. It has five equilateral triangular faces meeting at each vertex. It is represented by its Schläfli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 35. It is the dual of the regular dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. In most contexts, the unqualified use of the word "icosahedron" refers specifically to this figure. A regular icosahedron is a strictly convex deltahedron and a gyroelongated pentagonal bipyramid and a biaugmented pentagonal antiprism in any of six orientations. The name comes . The plural can be either "icosahedrons" or "icosahedra" (-drə). If the edge length of a regular icosahedron is , the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is while the midradius, which touches the middle of each edge, is where is the golden ratio. The surface area and the volume of a regular icosahedron of edge length are: The latter is F = 20 times the volume of a general tetrahedron with apex at the center of the inscribed sphere, where the volume of the tetrahedron is one third times the base area times its height . The volume filling factor of the circumscribed sphere is: compared to 66.49% for a dodecahedron. A sphere inscribed in an icosahedron will enclose 89.635% of its volume, compared to only 75.47% for a dodecahedron. The midsphere of an icosahedron will have a volume 1.01664 times the volume of the icosahedron, which is by far the closest similarity in volume of any platonic solid with its midsphere. This arguably makes the icosahedron the "roundest" of the platonic solids. The vertices of an icosahedron centered at the origin with an edge length of 2 and a circumradius of are where is the golden ratio.