The Cube

 

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

 

8.1)     

 

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

 

d =

 

So we have then

 

 

 

and so

 

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

 

8.3)                 

 

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

 

8.4)                 

 

where r is the radius if the inscribed sphere and R is the radius of the circumscribed sphere.

 

 

 

 

 

Figure 8.1

 

Triangle showing the circumscribing radius, R, from the center of the cube to a vertex of the cube.