  in geometry, a regular icosahedron ( or ) is a convex polyhedron with 20 faces, 30 edges and 12 vertices. it is one of the five platonic solids. it has five equilateral triangular faces meeting at each vertex. it is represented by its schlÃ¤fli symbol {3,5}, or sometimes by its vertex figure as 3.3.3.3.3 or 35. it is the dual of the dodecahedron, which is represented by {5,3}, having three pentagonal faces around each vertex. a regular icosahedron is 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" (-). ==dimensions== thumb|net folding into icosahedron if the edge length of a regular icosahedron is a, the radius of a circumscribed sphere (one that touches the icosahedron at all vertices) is :r_u = \frac{a}{2} \sqrt{\varphi \sqrt{5}} = \frac{a}{4} \sqrt{10 +2\sqrt{5}} = a\sin\frac{2\pi}{5} \approx 0.9510565163 \cdot a  and the radius of an inscribed sphere (tangent to each of the icosahedron's faces) is :r_i = \frac{\varphi^2 a}{2 \sqrt{3}} = \frac{\sqrt{3}}{12} \left(3+ \sqrt{5} \right) a \approx 0.7557613141\cdot a  while the midradius, which touches the middle of each edge, is :r_m = \frac{a \varphi}{2} = \frac{1}{4} \left(1+\sqrt{5}\right) a = a\cos\frac{\pi}{5} \approx 0.80901699\cdot a  where Ï (also called Ï) is the golden ratio. ==area and volume== the surface area a and the volume v of a regular icosahedron of edge length a are: :a = 5\sqrt{3}a^2 \approx 8.66025404a^2,  :v = \frac{5}{12} (3+\sqrt5)a^3 \approx 2.18169499a^3.  the latter is  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 :f=v/(4 \pi r_u^3/3) = \frac{20(3+\surd 5)}{(2\surd 5+10)^{3/2}\pi}\approx 0.6054613829. ==cartesian coordinates== 200px|thumb|right|icosahedron vertices form three orthogonal golden rectangles the following cartesian coordinates define the vertices of an icosahedron with edge-length 2, centered at the origin: :(0, Â±1, Â±Ï) :(Â±1, Â±Ï, 0) :(Â±Ï, 0, Â±1) where Ï=\tfrac{1+\sqrt{5}}{2} is the golden ratio (also written Ï). note
Deltahedra Planar_graphs Platonic_solids