Finding the radius, **R,** of the circumscribing circle
is equivalent to finding the distance from the centroid of the triangle to one
of the vertices. Finding the radius, **r**,
of the inscribed circle is equivalent to finding the distance from the centroid
to the midpoint of one of the sides.

If each vertex is connected to the midpoint of the opposite side by a straight line, then the lines intersect at the centroid of the triangle.

From **Figure 3.1**, **R** is the distance from the
centroid to a vertex and **r** is the distance from the centroid to the
midpoint of a side.

We have the following relationships:

_{}

From the table of trigonometric functions of common angles, we have

_{}

Therefore we have

_{} = _{}

and so

**3.1)** _{} = _{}

where **a** is the length of a side and **r**
is the radius of the inscribed circle.

For **R** we have

_{} = _{}

and so

**3.2)** _{} = _{}

where **a** is the length of a side and **R**
is the radius of the circumscribing circle.

In order to find **a** in terms of **R** we just turn **Equation
3.2** around like this:

_{} so _{}

But _{} so we have

_{}

So we finally have **a** in terms of **R** as

**3.3)** _{}

where **a** is the length of a side and **R**
is the radius of the circumscribing circle.

To find **r** in terms of **R** recall that from **Equation
3.1** we have

_{}

Substituting the value of a just found in this equation gives us

_{}

And so we have

**3.4)** _{}

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

**Equilateral triangle.
All sides are equal and all angles are 60 degrees.**

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