An octahedron is a solid figure having 8 sides, **6
vertices**, 12 edges and 8 faces.

Each face is an equilateral triangle. The ‘base’ of an octahedron is a square.

To find the radius of the circumscribing sphere, we need only to compute the distance from the center of the square base to a vertex of the square.

**Figure 7.1** shows isosceles triangles formed in half
of the square base. The distance from
the center of the square to a vertex of the square is the length of the radius,
**R**, of the circumscribing sphere.

We know from **Equation 4.2) **that

**7.1)** R = _{}

where **a** is the length of a
side and **R** is the radius of the circumscribed sphere.

**Figure 7.2** shows one face of an octahedron whose
sides are equal to **a**.

From this figure, the slant height of the face, **h**,
can be calculated as follows:

_{}

So,

h = _{}

Continuing with **Figure 7.2**, the distance along the
slant height to the centroid of the triangle, **h _{1}**, can be
calculated.

_{}

So we finally have

_{}

In **Figure 7.3** we have a triangle formed in a plane
that is through the 'top' vertex of the octahedron, perpendicular to the square
base of the figure, and intersecting one side at its midpoint.

**h** is the slant height just calculated. **h _{1}**

From this figure, we can see that

_{}

So then

_{}

_{}

so _{}

And finally

**7.2)** _{}

It is interesting to calculate the angles a and b.

From **Figure 7.3** we have the following:

_{}

b = 90 – a = 35.26 degrees

To find **a** and **r** in terms of **R** we start
with **Equation 7.1**. This equation
gives

_{}

So, solving for **a** we get

_{} But we also
know that _{}

So we have

_{}

So finally we have

**7.3)** _{}

where **a** is the length of a
side and **R** is the radius of the circumscribed sphere.

To find **r** in terms of **R** consider **Equation
7.2**:

_{} Substituting
the value of **a** in **Equation 7.3** here gives

_{}

And so finally we have

**7.4)** _{}

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

Isosceles triangles formed in the square base of an octahedron.

Equilateral triangle that is one face of an octahedron.

** **

** **

** **

Right triangle in the plane through the top vertex of the octahedron and through the midpoint of one of the sides.

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