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

Published by Pearson
ISBN 10: 0-32193-104-1
ISBN 13: 978-0-32193-104-7

Appendix A - Review - A.2 Geometry Essentials - A.2 Assess Your Understanding - Page A20: 39


$$2 \pi~units^2$$

Work Step by Step

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: $$ A=(\sqrt{ 2})^2\pi=2 \pi$$
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