# ICOSAHEDRON

See the icosahedron rotating around the x, y, z axis.
 Properties of the Icosahedron Faces: 20 trianglesVertices: 12, each with 5 edges meeting Edges: 30Dihedral angle: 138°11' The Symmetry Surface Area The surface area of an icosahedron is made up of 20 regular triangles. Volume One way of calculating the volume: the octahedron can be divided into 20 tetrahedra. Each can be calculated using the pyramid method. To get the vertices, take a look at this picture: Each vertex of the icosahedron lies on the edge of octahedron, is divided into 2 lines with the golden mean ratio Ø. Another way to calculate the volume is divide the icosahedron into 3 parts. The top part: The angle between the two red lines on the pentagon is 360/5=72°. Let r = radius of the pentagon, a = length of edge, by the cosine law: a² = r² + r ² - 2r²cos(72°) The height of the pyramid is calculated by the Pythagoras theorem. h² = r² + a². The bottom part of the icosahedron is the same. The top view of the middle piece: V = [5 × (3 + √5) / 12]a³