Find Ratio **s/a** of Square Inscribed in a Semicircle

By Leighton McIntyre

Goal : To find s/a, given a square of side * s* inscribed in a semicircle, and

The diagram is as follows:

Construct a radius of circle by taking the distance from the midpoint of the diameter to the vertex of of the square that touches the circumference of the circle.

Notice here that the radius , given by the purple line (r) is the same distance as the half diameter along the horizontal given by the blue line (a) and one half of the red line (1/2 s) . Thus r = 1/2 s + a (Equation 1).

Now using the fact that there is a right triangle formed by the purple line r one side of the square s and one half of a side of the square (1/2 s) ,