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A snub disphenoid with edge length has a surface area: [8] =, the area of 12 equilateral triangles. Its volume can be calculated as the formula: [ 8 ] V ≈ 0.85949 a 3 . {\displaystyle V\approx 0.85949a^{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.
Let ξ ≈ 0.943 151 259 24 be the real zero of the cubic polynomial x 3 + 2x 2 − φ 2, where φ is the golden ratio.Let the point p be given by = (+ +). Let the rotation matrices M 1 and M 2 be given by = (), = ().
For a snub cube with edge length , its surface area and volume are: [5] = (+) = + (). The snub cube is an Archimedean solid , meaning it is a highly symmetric and semi-regular polyhedron, and two or more different regular polygonal faces meet in a vertex. [ 6 ]
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.
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 ...
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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 ...