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On the Difference of Areas

The problem below that was originally proposed by Joseph Kennedy in School Science and Mathematics (52, 162, February 1952) appeared as a quickie in the Mathematics Magazine (Vol. 26, No. 5 (May - Jun., 1953), p. 287):

  A circle of radius 15 intersects another circle, radius 20, at right angles. What is the difference of areas of the non-overlapping portions?

Solution

Copyright © 1996-2010 Alexander Bogomolny

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Solution

The fact that the circles meet at right angles is a red herring. The solution is trivial and does not take that fact into account:

Let X be the area of the intersection. Then the remaining portions of the two circles have the areas (π 20² - X) and (π 15² - X), with the difference

 (π 20² - X) - (π 15² - X) = π 20² - π 15²,

independent of X.

Indeed, it is not important that the two shapes be circles. The numerical answer will be the same for any two blobs of areas π 20² and π 15².

Copyright © 1996-2010 Alexander Bogomolny

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