$3$ . In the figure below, points $E, F, G$ , and $H$ are on the sides of square $A B C D$ . Arc \overparen{E H} has center at A, \overparen{E F} at B, \overparen{F G} at $C$ , and $\widehat{G H}$ at $D$ . All of the arcs have radius of $3$ feet. What is the area, in square feet, of th shaded region? $\newline$ A. $24-6 \pi$ $\newline$ B. $H$ $0$ $\newline$ C. $H$ $1$ $\newline$ D. $H$ $2$ $\newline$ E. $H$ $3$

Full solution

3

. In the figure below, points

E, F, G

, and

H

are on the sides of square

A B C D

. Arc \overparen{E H} has center at A, \overparen{E F} at B, \overparen{F G} at

C

, and

\widehat{G H}

D

. All of the arcs have radius of

3

feet. What is the area, in square feet, of th shaded region?

\newline

24-6 \pi

\newline

H

0

\newline

H

1

\newline

H

2

\newline

H

3

Calculate Side Length: The area of the square is the side length squared. Since the radius of the arcs is $3$ feet, the side length of the square is twice the radius, which is $6$ feet. $\newline$ Area of square = side length $^2$ = $6^2$ = $36$ square feet.
Find Area of Quarter-Circle: Each arc forms a quarter-circle with radius $3$ feet. The area of a full circle with radius $3$ feet is $\pi r^2 = \pi(3)^2 = 9\pi$ square feet.
Calculate Total Shaded Area: Since each arc is a quarter-circle, the area of each shaded quarter-circle is $\frac{1}{4}$ of the full circle's area. $\newline$ Area of each shaded quarter-circle = $\left(\frac{1}{4}\right) \times 9\pi = \left(\frac{9}{4}\right)\pi$ square feet.
Find Area of Non-Shaded Region: There are $4$ shaded quarter-circles in total. To find the total area of the shaded regions, multiply the area of one quarter-circle by $4$ . $\newline$ Total shaded area = $4 \times \left(\frac{9}{4}\right)\pi = 9\pi$ square feet.
Find Area of Non-Shaded Region: There are four shaded quarter-circles in total. To find the total area of the shaded regions, multiply the area of one quarter-circle by $4$ . $\newline$ Total shaded area = $4 \times \left(\frac{9}{4}\right)\pi = 9\pi$ square feet. Subtract the total shaded area from the area of the square to find the area of the non-shaded region. $\newline$ Area of non-shaded region = Area of square - Total shaded area = $36 - 9\pi$ square feet.