Happiness is Playing With CirclesAugust 17, 2014
When I saw this puzzle by Curmudgeon, I was seriously intrigued.
After doing some algebra, I figured out that the area of the regions had to be , 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 . 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.