## 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?

Source: NCTM Mathematics Teacher, January 2006

SOLUTION

$a^2+(a/2)^2=r^2$
$4a^2+a^2=4r^2$
$5a^2=4r^2$
$a^2=4r^2/5$
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: $\approx 74.5\%$