The 'Square Circle' Puzzle (Part 2)

I have outlined one way to solve the 'Square Circle' puzzle originally posed in my last post.

Before jumping into the solution, I want to share the origins of the bizarre expression used frequently in the worlds of boxing and wrestling.  In short, boxing rings were circular centuries ago.  In 1838, the first square ring was introduced (thanks Wikipedia!).  So, despite the fact that modern boxing rings are square, they are often referred to as square circles for historical reasons.

Now, what ratio of the identical areas of concentric square and circle shapes is shared?  One straight forward method is shown below:

If we cut out a specific fraction of the circle on the left, we can determine the area of the purple section, and the rest is relatively simple.  The key is identifying the relationship between the parameters 'a' and 'r'.  Consider a circle with radius 'r' and a square with each side measuring '2a'.  If their areas are equivalent, then 4a^2 = pi*r^2.

We can then express 'b' in terms of 'a' and 'r' using the Pythagorus rule for right angle triangles and express it entirely in terms of 'a'.  The area of the triangle can then be easily found.

To find the area of the entire disc portion, we must first find angle theta in terms of 'a' and 'r' (use the 'cah' in 'sohcahtoa').  Then, the area of the disc is a known ratio of the total circle.

Finally, the purple shaded area is the difference between the disc portion and the triangular portion.  The value of this area, in terms of 'a' is:

The total 'not-shared' area is a collection of 8 of these purple areas.  As an interesting side note, you may have noticed that each white section on the left image above has the same area due to the symmetry of the situation.

Then the shared area is the difference between the total area and the 'not-shared' area.  If we take the ratio of shared vs total, we arrive at the solution to the posed question.  The exact value is given by:

If you have solved the problem using a different method, you may have a different-looking solution that is also correct.  Note that, as expected, the ratio is independent of the size of the shapes.  If we approximate the above expression in decimal form, we see that about 91% of the area of the concentric shapes is shared, which looks about right.  Though it serves no practical purpose, arriving at this result was fun - certainly more fun than watching most boxing matches.