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The surface area and the volume of the truncated icosahedron of edge length are: [2] = (+ +) = +. The sphericity of a polyhedron describes how closely a polyhedron resembles a sphere. It can be defined as the ratio of the surface area of a sphere with the same volume to the polyhedron's surface area, from which the value is between 0 and 1.
Animated truncation sequence from {5 ⁄ 2, 5} to {5, 5 ⁄ 2}Its convex hull is the regular convex icosahedron.It also shares its edges with the great icosahedron; the compound with both is the great complex icosidodecahedron.
The surface area A and the volume V of the rhombic dodecahedron with edge length a are: [4] =, =. The rhombic dodecahedron can be viewed as the convex hull of the union of the vertices of a cube and an octahedron where the edges intersect perpendicularly.
The volume of a rhombicuboctahedron can be determined by slicing it into two square cupolas and one octagonal prism. Given that the edge length a {\displaystyle a} , its surface area and volume is: [ 7 ] A = ( 18 + 2 3 ) a 2 ≈ 21.464 a 2 , V = 12 + 10 2 3 a 3 ≈ 8.714 a 3 . {\displaystyle {\begin{aligned}A&=\left(18+2{\sqrt {3}}\right)a^{2 ...
If the edge length of a regular dodecahedron is , the radius of a circumscribed sphere (one that touches the regular dodecahedron at all vertices), the radius of an inscribed sphere (tangent to each of the regular dodecahedron's faces), and the midradius (one that touches the middle of each edge) are: [21] =, =, =. Given a regular dodecahedron ...
The volume of a tetrahedron can be obtained in many ways. It can be given by using the formula of the pyramid's volume: =. where is the base' area and is the height from the base to the apex. This applies for each of the four choices of the base, so the distances from the apices to the opposite faces are inversely proportional to the areas of ...
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Doubling the cube is the construction, using only a straightedge and compass, of the edge of a cube that has twice the volume of a cube with a given edge. This is impossible because the cube root of 2, though algebraic, cannot be computed from integers by addition, subtraction, multiplication, division, and taking square roots.
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