Answer
total area of shaded region = area of circle - area of square = 2$\pi$ - 4
Work Step by Step
The total area of the shaded region = area of circle - area of square
Since we are given the side length of the square, we can use the Pythagorean Theorem to calculate the diagonal of the square, which will then give us the diameter/radius of the circle.
Recall that the Pythagorean Theorem states that $a^{2}$ + $b^{2}$ = $c^{2}$, where a and b are the side lengths and c is the hypotenuse.
SOLVING FOR THE DIAGONAL OF THE SQUARE:
$a^{2}$ + $b^{2}$ = $c^{2}$
$2^{2}$ + $2^{2}$ = $c^{2}$
8 = $c^{2}$
c = $\sqrt 8$ = 2$\sqrt 2$
The diagonal of the square is equal to 2$\sqrt 2$
SOLVING FOR THE RADIUS OF THE CIRCLE:
We know that the diagonal of the square is 2$\sqrt 2$ which is also equal to the diameter of the circle. This means that we can divide 2$\sqrt 2$ by 2 to get the radius of the circle.
Radius of the circle = 2$\sqrt 2$$\div$2 = $\sqrt 2$
SOLVING FOR THE AREA OF THE CIRCLE:
Recall that the area of a circle is $\pi$$r^{2}$
Area of the circle
=$\pi$$r^{2}$
=$\pi$$\sqrt 2$$^{2}$
=2$\pi$
The total area of the circle is 2$\pi$
SOLVING FOR THE AREA OF THE SQUARE:
The area of a square is equal to $s^{2}$, where s is the side length of the square
Area of square = $s^{2}$ = $2^{2}$ = 4
The area of the square is 4
SOLVING FOR THE AREA OF THE SHADED REGION:
area of shaded region = area of circle - area of square
= 2$\pi$ - 4
The area of the shaded region is equal to 2$\pi$ - 4
