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Therefore, the circumradius of this rhombicosidodecahedron is the common distance of these points from the origin, namely √ φ 6 +2 = √ 8φ+7 for edge length 2. For unit edge length, R must be halved, giving R = √ 8φ+7 / 2 = √ 11+4 √ 5 / 2 ≈ 2.233.
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 ...
Considering that each length of the regular octahedron is , and the edge length of a square pyramid is (the square pyramid is an equilateral, the first Johnson solid). From the equilateral square pyramid's property, its volume is 2 6 a 3 {\textstyle {\tfrac {\sqrt {2}}{6}}a^{3}} .
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.
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 ...
The 600-cell also contains unit-edge-length cubes and unit-edge-length octahedra as interior features formed by its unit-length chords. In the unit-radius 120-cell (another regular 4-polytope which is both the dual of the 600-cell and a compound of 5 600-cells) we find all three kinds of inscribed icosahedra (in a dodecahedron, in an octahedron ...
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 long radius (center to vertex) of the icosidodecahedron is in the golden ratio to its edge length; thus its radius is φ if its edge length is 1, and its edge length is 1 / φ if its radius is 1. [4]