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
|Cube||6||4||2 / √3|
|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|
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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