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© Copyright 2002, Jim Loy
On the left is a blue crescent
made up of a semicircle and a quarter circle (of a larger circle). If the width
of the figure (diameter of the semicircle) is 1, what is the area of the
crescent? Try to figure it out, before looking at the solution below.
Solution: The area of the semicircle is pi/8 (pi r^2, where ^2 means squared). The area of the quarter circle (with radius sqrt(2)/2 by The Pythagorean Theorem, where sqrt() is the square root function) is pi/8, a coincidence? The area of the green portion of this quarter circle is pi/8-1/4. The area of the blue crescent is therefore pi/8-pi/8+1/4 = 1/4. In other words, the blue crescent has the same area as the yellow triangle (blue+green=green+yellow, so blue=yellow, regardless of the units). It may surprise you that an area completely bounded by curves is equal to such a simple number. This was discovered by Hippocrates.