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

From **Figure 4.1**,
**R** is the distance from the center to a vertex and **r** is the
distance from the center 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 finally

**4.1)** _{}

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

For R we have

_{}

And so finally

**4.2)** _{}

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

To find **a** and **r** in terms of **R** we start
with **Equation 5.2**. This equation
gives

_{}

So, solving for **a** we get

_{} But we also
know that _{}

So we have

_{}

So finally we have

**4.3)** _{}

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

To find **r** in terms of **R** consider **Equation
4.1**:

_{} Substituting
the value for **a** in **Equation 4.3** gives us

**4.4)** _{}

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

**The square. All sides are equal and all angles are 90
degrees.**

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