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.

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

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