## Precalculus: Concepts Through Functions, A Unit Circle Approach to Trigonometry (3rd Edition)

$$2 \pi~units^2$$
The shaded region is a circle. To find the radius, we use Thales's theorem, which states that if we have distinct points A, B, C on a circle and AC is the diameter, then we have a 90 degree (right) angle at B. In other words, the shape inside the circle is a square with right angles. Hence, by applying the Pythagorean theorem with the sides $a=b=2$, we can find the diameter (hypotenuse): $c^2=a^2+b^2=2^2+2^2=8$ $c=\sqrt{8}=2\sqrt{2}$ Thus, the radius is $c/2=\sqrt{2}$. Now, we can calculate the area of the circle as: $$A=(\sqrt{ 2})^2\pi=2 \pi$$