## Overlapping Circles

A circle of radius 15 inches intersects another circle, radius 20 inches, at right angle (see below). What is the difference of the areas of the non-overlapping portions?

Source: Julia Robinson Mathematics Festival

SOLUTION
Let $x$ be the area of the overlap portion.

Area of the non-overlapping portion of the big circle = area of big circle – area of overlap
$=\pi 20^2-x$

Area of the non-overlapping portion of the small circle = area of small circle – area of overlap
$=\pi 15^2-x$

Difference of the areas of the non-overlapping portions is
$\left (\pi 20^2-x\right )-\left (\pi 15^2-x\right )=\pi 20^2-x-\pi 15^2+x$
$=\pi 20^2-\pi 15^2$
$=\pi \left (20^2-15^2\right )$
$=\pi \left (20+15\right )\left (20-15\right )$
$=\pi \left (35\right )\left (5\right )$
$=175\pi$

Answer: $175\pi$ square inches.

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## About mvtrinh

Retired high school math teacher.
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