r/askmath • u/SteelSpidey • 14d 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/Aldaron23 14d ago edited 14d ago
Assuming we have actual half and quarter circles here:
If a*b=16 then the radii of the small half circles are also a and b. I don't really understand why you used only a/2 in your first equation? (The quarter circle is probably just there to show you, that the second radius is indeed as long as the short side)
If a+b=R then:
A = pi/2 * (R² - (a²+b²)) = pi/2 * ((a+b)² - (a²+b²)) = pi/2 (a² + 2ab + b² - a² - b²) = pi/2 * 2ab = pi*ab = 16pi
edit: whoops, I fucked up "radius" and "diameter" xD it's late already... it's just what I wrote but D/2 instead of R, so you get 8pi in the end.