The cube is composed of six square faces that meet each other at right angles( 90 degrees). It has eight vertices and 12 edges.
In a cube with sides of length a the radius of the inscribed sphere is equal to the radius of the inscribed circle for any of the square faces. From Equation 4.1) we know that
To find the radius, R, of the circumscribing sphere, we must find the distance from the center of the cube to any vertex.
From Figure 8.1 we have that
where d is the circumscribing radius of a square side.
From Equation 4.2) we know that
So we have then
8.2) R =
To find a and r in terms of R we start with Equation 8.2. This equation gives
So, solving for a we get
But we also know that
So we have
So finally we have
where a is the length of an edge and R is the radius of the circumscribed sphere.
To find r in terms of R consider Equation 8.1:
Substituting the value for a in Equation 8.3 gives us
where r is the radius if the inscribed sphere and R is the radius of the circumscribed sphere.
Triangle showing the circumscribing radius, R, from the center of the cube to a vertex of the cube.