Square in Circle

A square of maximum area is inscribed in a semicircle as shown. What percent of the area of the circle is outside the square?
image
Source: NCTM Mathematics Teacher, January 2006

SOLUTION
image
a^2+(a/2)^2=r^2
4a^2+a^2=4r^2
5a^2=4r^2
a^2=4r^2/5
area of circle = \pi r^2
area of square = a^2
area of circle outside of square
\pi r^2-4r^2/5=r^2(5\pi-4)/5
ratio of area of circle outside of square to area of circle
r^2(5\pi-4)/5 \div \pi r^2\approx .7453
percent of the circle outside of square \approx 74.5\%

Answer: \approx 74.5\%

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About mvtrinh

Retired high school math teacher.
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