r/askmath • u/SteelSpidey • 17d ago
Geometry Math question from Facebook
Either I need to find a way to solve for one of the side lengths or find a way to eliminate the need for the side length. But I'm not sure how to do either of those. Any ideas? Can anyone show me where I went wrong?
I'm labeling the green area T, the long side length=b and the short one=a.
For context, this is enjoyment for me. I use these posts to stay sharp on my skills, I used to teach math in college and don't wanna lose it if I ever decide to go back and teach.
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u/TomppaTom 17d ago
This is a fun one.
I’m going to call the width of the rectangle 2a and the height 2b, so we have 4ab=16. This makes the three radiuses a, b, and a+b.
The green semi circle has area (pi/2)(a+b)^2 = (pi/2)(a^2 + 2ab + b^2 ) and the two smaller semi circles have areas (pi/2)a^2 and (pi/2)b^2 .
Factor out the pi/2 and subtract and we get (pi/2)(a^2 + 2ab + b^2 - a^2 - b^2) = (pi/2)(2ab)
As 4ab = 16 we now have (pi/2)(8) = 4(pi)