Square inscribed in a semicircle

by Hwa Young Lee

Final assignment 2. Square Inscribed in a Semicircle: Find a Ratio

This is a GSP construction of this figure.

Given a semicircle with an inscribed square of side s, let a be the length on the diameter on each side of the square.

Let's find the ratio of the length of the side of the square to the length on the diameter on each side of the square.

In other words, let's find .

We want to use the fact that we have a semicircle and square. If we add a magic line, it is pretty easy to find the ratio:

First of all, we know that the midpoint of the bottom side of the square is the center of the semicircle.

Hence, if we let the radius of the semicircle r, then we know that

Also, we have a right triangle formed by the magic line and the side of the square and half the side of the square.

We use the Pythagorean Theorem and have

Plugging in , we have

With a little expanding and canceling out, we are left with

Dividing out by (a is nonzero),

Substituting as x, all we need to do is solve

and find the value of our ratio!
Using the quadratic formula, we get

but since a and s are both positive (lengths), our wanted ratio is

the golden ratio!
We earlier found the golden ratio in the Fibonacci sequence using the spreadsheet in assignment 12.