Harmonic Ratio of the Cube

Let D and R be the diameter and radius of the circumscribed sphere, S and s the side and semi-side of the cube, and d and r the diameter and radius of the inscribed sphere. Then according to Euclid Book XIII, Prop. 15, D = sqrt(3)*S, or
[1] R = sqrt(3)*s
Our goal is to relate R and r, so we proceed to relate s and r. It is obvious in this case that s = r, so
[2] R = sqrt(3)*r
but we wish to establish a simple method of discovering the relation between s and r, that will be useful later.

The central triangle argument for the cube

Consider the central triangle: center of the cube to any vertex, then to the center of an adjacent face, making a right triangle. The sides are: center of cube to vertex = R, center of cube to center of face = r, and closing up within the face, c. Thus, by the Pythagorean theorem, R2 = r2 + c2. As the face is a square, we know that c = sqrt(2)*s, so
[3] R2 = r2 + 2*s2
and from [1], s2 = R2/3, so
[4] R2 = r2 + (2/3)*R2,
or R = sqrt(3)*r.
Rev'd 18 jan 2002 by Ralph Abraham