Geometry Problem Involving Two Circles (See Comments for Solution)

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?

two circles

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.

About RobertLovesPi

I go by RobertLovesPi on-line, and am interested in many things. Welcome to my little slice of the Internet. The viewpoints and opinions expressed on this website are my own. They should not be confused with the views of my employer, nor any other organization, nor institution, of any kind.
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2 Responses to Geometry Problem Involving Two Circles (See Comments for Solution)

  1. howardat58 says:

    Two equilateral triangles, cosine rule.
    If you want the proportion of the area then you are on your own!!!

    Liked by 1 person

  2. Robert Webb says:

    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.

    Liked by 1 person

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