This is a puzzle I made up not long ago. After trying to solve it for a bit (no success yet, but I haven’t given up), I decided to share the fun.

A small circle of radius r is centered on a large circle of radius R. It is a given that 0 < r < R. In terms of r and R, what fraction of the smaller circle’s circumference lies outside the larger circle?

I am 90% certain there is an extremely simple way to do this, using only things I already know. It’s frustrating that the answer isn’t simply leaping out of the computer screen, at me. For simple math problems, that’s what usually happens . . . so either this is merely deceptively simple, or I am missing something.

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Two equilateral triangles, cosine rule.

If you want the proportion of the area then you are on your own!!!

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I don’t see where equilateral triangles come into it, nor the cosine rule for that matter. I did get an answer though. I get this fraction:

1 – angle / pi

where

angle = acos( r / (2R) )

acos() being the inverse of cos() in radians.

A quick test of the trivial case where r = R gives the correct answer of 2/3.

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