The Square

 

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.

 

 

 

Figure 4.1

 

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