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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 = pi / 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 sqrt(4sin2a - s2) / (2sin a). Hence its volume is fks2sqrt(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 = (sqrt(5) - 1) / 2 = 0.618034 is the golden ratio then we have

  fks
Tetrahedron 43sqrt(8/3)
Cube 642 / sqrt(3)
Octahedron 83sqrt(2)
Dodecahedron 1252 / sqrt(5 + 4p + p2)
Icosahedron 2032p / sqrt(1 + p2)

We can also calculate the volumes for the rhombic dodecahedron (2 and 4sqrt(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
Tetrahedron64 0.51320012.3% 13.856420330.8%
Cube86 1.53960136.8% 8.000000191.0%
Octahedron68 1.33333331.8% 6.928203165.4%
Rhombic dodecahedron1412 2.00000047.7% 5.656854135.0%
Dodecahedron2012 2.78516466.5% 5.550290132.5%
Icosahedron1220 2.53615160.5% 5.054058120.7%

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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