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Question from Scott, a student:

There are two circles, big circle with radius R and small one with radius r. They intersect and overlap in such a way that the common area formed is 1/2 pi r^2 (half the area of the small circle). The Question is: suppose we have known the radius r of the small circle, and the distance between the two circle centers, what should the radius R of the large circle be?

Scott,

My guess is that you are looking for a formula for R, in which case you are out of luck. To solve your problem you must obtain an equation in the unknown radius R by equating the areas of the sectors of two circles (which means twice using formula (18) of mathworld:
http://mathworld.wolfram.com/CircularSegment.html,
where the "h" in that formula each time is itself an expression involving d, r, and R).
Let us know if you would like to see the details. Once you get the equation, you solve it for R using standard computer techniques. The problem is not very hard, but it is certainly not suitable for a high-school geometry class.

Chris

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