In the figure below, points $E,F,G$ , and $H$ are on the sides of square $ABCD$ . Arc $\widehat{EH}$ has center at $A$ , $\widehat{EF}$ at $B$ , $\widehat{FG}$ at $C$ , and $\widehat{GH}$ at $H$ $0$ . All of the arcs have radius of $H$ $1$ feet. What is the area, in square feet, of th shaded region? $\newline$ A. $H$ $2$ $\newline$ B. $H$ $3$ $\newline$ C. $H$ $4$ $\newline$ D. $H$ $5$ $\newline$ E. $$36$$-9$\pi

Full solution

Q. In the figure below, points

E,F,G

, and

H

are on the sides of square

ABCD

. Arc

\widehat{EH}

has center at

A

\widehat{EF}

B

\widehat{FG}

C

, and

\widehat{GH}

H

0

. All of the arcs have radius of

H

1

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

\newline

H

2

\newline

H

3

\newline

H

4

\newline

H

5

\newline

E. $$36$$-9$\pi

Calculate Square Area: The area of the square is side length squared. Since the radius of the arcs is $3$ feet, the side length of the square is $6$ feet. $\newline$ Area of square = $\text{side}^2 = 6^2 = 36$ square feet.
Calculate Arc Area: Each arc forms a quarter of a 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 Arc Area: Since each arc is a quarter of a circle, the area of each arc is $\frac{1}{4}$ of the area of a full circle. $\newline$ Area of each arc = $(\frac{1}{4}) \times 9\pi = (\frac{9}{4})\pi$ square feet.
Calculate Total Shaded Area: There are four arcs, so the total area of the arcs is $4$ times the area of one arc. $\newline$ Total area of arcs = $4 \times \left(\frac{9}{4}\right)\pi = 9\pi$ square feet.
Calculate Total Shaded Area: There are four arcs, so the total area of the arcs is $4$ times the area of one arc. $\newline$ Total area of arcs = $4 \times \left(\frac{9}{4}\right)\pi = 9\pi$ square feet.The shaded region is the area of the square minus the total area of the arcs. $\newline$ Area of shaded region = Area of square - Total area of arcs = $36 - 9\pi$ square feet.