Arcs in a Square (or Snakes on Plane)

In a recent MAA publication, Shai Simonson, attempts to bring the joys and excitement of the world of mathematics to the non-technical reader.  In Rediscovering Mathematics: You Do the Math, Simonson covers a wide array of topics ranging from number theory to the application of probability in sports, casinos and gambling.  I have added the book to my reading list and you might want to take a look at the article that begins his text, “How to Read Mathematics.”

One of the problems from the book was posted over at Math Mama Writes… and when a puzzle like this piques my interest, I’m at its mercy until I figure it out.  Thanks to a recent illustration I made in calculus last week and a cartoon that reinforced my perspective, I have a new motto for next year’s courses:

Math problems aren’t solved, they are conquered!

Well, this problem below was one that I had to defeat.  I went to battle with it and after losing a few skirmishes (i.e., trying approaches that failed) I finally beat it into submission.  From now on, when I see that feisty integral that won’t behave or a simple number puzzle whose pattern defies identification, I’ll strap on my armor (or sweater vest), grab my sword (or calculator) and wage full-out war on that problem.  No problem is safe!

Arcs in a Square (or Snakes on a Plane)

Given the square ABCD, with side length 4 and circular arcs centered at each vertex, find the area of the region at the center – without using calculus.

And by the way, the Snakes on a Plane reference is my own and I’ve not ever heard this problem referred to in this way but it sounded good to me (arcs [tex]approx[/tex] snakes, square [tex]approx[/tex] plane).  However, I am a strange duck.

I’ll post a couple of different approaches that conquered this problem in later posts.