How to Solve this Square-Circle-Square Problem

Having some fun with geometry and algebra

Picture a circle with both its inner and outer square:

The yellow line in the picture intersects with the top left corners of both squares and the center of the circle (and the bottom right corners of both squares). The length of the continuous (non-dotted) line segment is 1.

What is the area of the inner (blue) square?

You may be surprised to find that we can calculate the exact area of the inner square with just that one line segment to go on, but we can.

I encourage the reader to give this problem a try first!

Done?

How to find the area of the inner square

First, let’s call the radius of the circle r and draw it two times like this:

The partially dotted line segment from the center to the upper left corner of the outer square has length r + 1. We can know use the Pythagorean theorem to find the value of r: