Dan Meyer recently posted a nice geometry and algebra problem on his blog for discussion. Here is the original text of the problem:

Given an arbitrary point P on a line segment AB, let AP form the perimeter of a square and PB form the circumference of a circle. Find P such that the area of the square and circle are equal.

A lot has already been written on that problem on his blog and others. I wanted to post solutions to two interesting extensions of the problem in case anyone was interested.

The first solution is for dividing the line into an n-sided regular polygon and a circle:

The other solution is for dividing the line into two different n-sided regular polygons:

It’s worth noting that most people (me included at first) seem to solve these problems using quadratic equations, but that is not necessary (which is convenient since my students are not yet able to solve quadratics). There is nice way to solve them by just setting up a simple fraction based on the perimeters and setting the areas equal.

It was fun to check the limit as n went to infinity to see that I did end up with the circle solutions as well.