**Exploring
Inscribed Squares in a Half Circle **

By: Amanda Sawyer

For the following figure, we will determine the ratio of the
side length of the inscribed square (** s**) and the additional distance of the square to the circle (

The picture below shows the labeled areas of our figure.

We can already determine certain relationship between ** s** and

This now allows us to determine where the center of our circle occurs.

We can notice that it is located directly in the middle of
side** s**. Since we have equal distance of

In the inscribed square, we can find the radius located from the center of our circle to one of its sides as seen below.

This radius creates a triangle within our shape with a right
angle. Therefore, we can use the Pythagorean
Theorem to determine a relationship between ** s**
and

When we simplify the left and right side of the equation, we obtain:

By using the quadratic equation, we can solve for ** s.** From this value of

This gives us the two different ratios for the values. However, we know that both ** s**
and

We know that this ratio is phi (the golden ratio) which I discussed in detail in Final 3 assignment.