Happiness is Playing With Circles

August 17, 2014

When I saw this puzzle by Curmudgeon, I was seriously intrigued.

The circles have radius 1, and the seven lettered regions are of equal area. The question is, what is the area of the pentagon?

After doing some algebra, I figured out that the area of the regions had to be \frac{\pi}{4}, but the problem is figuring out the height of the pentagon. Not knowing what else to do, I set it up in Geometer’s Sketchpad. I’m trying to get used to using Geogebra, but I figured I would work it out first in Sketchpad. I’ll put my solution after the “more” in case you want to work it out for yourself.

My biggest complaint with Sketchpad (and Geogebra, from what I’ve seen) is that I can’t just set a length as a unit of 1 — everything has to measure out in real units. This meant I had to come up with a way to set a unit length, and then convert all of the measurements from square centimeters to square units. As best as I can tell, the area of the pentagon is \pi. I haven’t worked it out algebraically, but here’s how it looks geometrically:


I worked out the area several hours ago. Since then, I’ve been playing with automating and then animating changing the size. The trickiest part was figuring out the relationship between the radius and the height of the pentagon (the bottom side angles are about 99.6°, in case you were curious). Once I got the distance between the centers and the height of the pentagon linked together, it took a while to link my “slider” with everything.

I’ve had the “Happy” song going through my head for the last hour or so. This was a fun way to spend my last Saturday before going back to school. There’s some actual application to my classes, but it’s way too late to go through that right now. I was supposed to be in bed about 3 hours ago.



  1. If each region is equal in area (pi/4), and the pentagon is made up of four of these regions then the pentagon’s area would be 4*(pi/4). Pi!

    • Cool! Yeah, that does seem very obvious, now that you mention it! I think I got confused off the bat by not seeing region G — I got it mixed up with the pentagon, and it wasn’t until I was halfway through making my diagram that I realized what G was.

      In all honesty, most of my time was spent making everything look pretty and move together. It made for a nice review of Sketchpad that I can use to play with Geogebra. Circles are our friends!

  2. I LOVE this problem!!! I will be putting it up on the wall at school and giving a prize to the first kid who can solve it!

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