Regular icosahedron
Mensuration

The insphere of a convex polyhedron is a sphere inside the polyhedron, touching every face. The circumsphere of a convex polyhedron is a sphere that contains the polyhedron and touches every vertex. The midsphere of a convex polyhedron is a sphere tangent to every edge. Therefore, given that the edge length a {\displaystyle a} of a regular icosahedron, the radius of insphere (inradius) r I {\displaystyle r_{I}}, the radius of circumsphere (circumradius) r C {\displaystyle r_{C}}, and the radius of midsphere (midradius) r M {\displaystyle r_{M}} are, respectively: r I = φ 2 a 2 3 ≈ 0.756 a, r C = φ 2 + 1 2 a ≈ 0.951 a, r M = φ 2 a ≈ 0.809 a. {\displaystyle r_{I}={\frac {\varphi ^{2}a}{2{\sqrt {3}}}}\approx 0.756a,\qquad r_{C}={\frac {\sqrt {\varphi ^{2}+1}}{2}}a\approx 0.951a,\qquad r_{M}={\frac {\varphi }{2}}a\approx 0.809a.}

The surface area of polyhedra is the sum of its every face. Therefore, the surface area of regular icosahedra A {\displaystyle A} equals the area of 20 equilateral triangles. The volume of a regular icosahedron V {\displaystyle V} is obtained by calculating the volume of all pyramids with the base of triangular faces and the height with the distance from a triangular face's centroid to the center inside the regular icosahedron, the circumradius of a regular icosahedron; alternatively, it can be ascertained by slicing it off into two regular pentagonal pyramids and a pentagonal antiprism, and adding up their volume. The expressions of both are: A = 5 3 a 2 ≈ 8.660 a 2, V = 5 φ 2 6 a 3 ≈ 2.182 a 3. {\displaystyle A=5{\sqrt {3}}a^{2}\approx 8.660a^{2},\qquad V={\frac {5\varphi ^{2}}{6}}a^{3}\approx 2.182a^{3}.} A problem dating back to the ancient Greeks is determining which of two shapes has a larger volume, an icosahedron inscribed in a sphere, or a dodecahedron inscribed in the same sphere. The problem was solved by Hero, Pappus, and Fibonacci, among others. Apollonius of Perga discovered the curious result that the ratio of volumes of these two shapes is the same as the ratio of their surface areas. Both volumes have formulas involving the golden ratio, but taken to different powers. As it turns out, the icosahedron occupies less of the sphere's volume (60.54%) than the dodecahedron (66.49%).

The dihedral angle of a regular icosahedron can be calculated by adding the angle of pentagonal pyramids with regular faces and a pentagonal antiprism. The dihedral angle of a pentagonal antiprism and pentagonal pyramid between two adjacent triangular faces is approximately 38.2 °. The dihedral angle of a pentagonal antiprism between pentagon-to-triangle is 100.8 °, and the dihedral angle of a pentagonal pyramid between the same faces is 37.4 °. Therefore, for the regular icosahedron, the dihedral angle between two adjacent triangles, on the edge where the pentagonal pyramid and pentagonal antiprism are attached is 37.4 ° + 100.8 ° = 138.2 °.
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