# Square inside a semicircle

## Problem

You are given a semicircle with an inscribed square of side length $$s.$$ Let $$a$$ be the length of the diameter on either side of the square. Find the value of $$s / a.$$

## Solution

Right away, you should notice that $$a$$ and $$s$$ are both dependant on the radius of the semicircle, let's call it $$r.$$ Because we're looking for the value of $$s / a,$$ we should start by looking for ways to express $$r$$ in terms of $$s,$$ $$a,$$ or both.

We notice that there is a side of the square located on the diameter of the semicircle. Since the length of the diameters are equal on both sides of side s on the diameter, we can conclude that the midpoint on that particular side s is also the midpoint of the diameter of the circle. So the radius r of the semicircle must be $$r = (1/2)s + a.$$

We also know that a segment formed from the midpoint of the semicircle to the upper right vertex of the square is also the radius of the semicircle. Therefore, it is possible to form a right triangle, as seen here:

Now we can use the Pythagorean theorem to express $$r$$ in another way:

\begin{align} r^2 &= s^2 + \left(\dfrac{1}{2}s\right)^2 \\[1em] r^2 &= s^2 + \dfrac{1}{4}s^2 \\[1em] r^2 &= \dfrac{5}{4}s^2 \\[1em] r &= \dfrac{\sqrt{5}}{2}s \end{align}

Now we have two ways of expressing $$r,$$ so let's set them equal to one another:

$$r = \dfrac{\sqrt{5}}{2}s = \dfrac{1}{2}s + a$$

Now let's try solving for $$s / a:$$

\begin{align} a &= \dfrac{\sqrt{5}}{2}s - \dfrac{1}{2}s \\[1em] a &= \left(\dfrac{\sqrt{5 - 1}}{2}\right)s \\[1em] \dfrac{a}{s} &= \dfrac{\sqrt{5 - 1}}{2} \\[1em] \dfrac{s}{a} &= \dfrac{2}{\sqrt{5 - 1}} \end{align}