# Platonic solids inside and outside a unit sphere

Platonic solids inside and outside a unit sphere

Q: What are the volumes of the platonic solids inscribed in/circumscribed about a unit sphere?
[TB, 11 Apr 1996]

A: Consider a given solid as a set of pyramids on the faces. Suppose there are f faces each with k sides of length d, and let a = π / k. The area of the pyramid's base, ie the face, is k(s/2)2 / tan a. For the inscribed solid the pyramid's slant height (measured from a vertex of the base) is 1, so its height measured normal to the base is √(4sin2a - s2) / (2sin a). Hence its volume is fks2√(4sin2a - s2) / (24sin a tan a).

For the circumscribed solid we can just divide the volume of the inscribed solid by the cube of the above normal height.

Now if p = (√5 - 1) / 2 = 0.618034 is the golden ratio then we have

 f k s Tetrahedron 4 3 √(8/3) Cube 6 4 2 / √3 Octahedron 8 3 √2 Dodecahedron 12 5 2 / √(5 + 4p + p2) Icosahedron 20 3 2p / √(1 + p2)

We can also calculate the volumes for the rhombic dodecahedron (2 and 4√2) quite easily. Thus we have the following - percentages indicate volume relative to that of the sphere, ie 4.188790:

 # vertices # faces Volume of inscribed solid Volume of circumscribed solid Tetrahedron 6 4 0.513200 12.3% 13.856420 330.8% Cube 8 6 1.539601 36.8% 8.000000 191.0% Octahedron 6 8 1.333333 31.8% 6.928203 165.4% Rhombic dodecahedron 14 12 2.000000 47.7% 5.656854 135.0% Dodecahedron 20 12 2.785164 66.5% 5.550290 132.5% Icosahedron 12 20 2.536151 60.5% 5.054058 120.7%

Among the regular solids the inscribed solids' volumes are in order of # vertices and the circumscribed solids' are in order of # faces [David Cartwright].

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