Question

the circle has center o. its radius is 8 in, and the central angle a measures 90°. what is the area of the shaded region? give the exact answer in terms of π, and be sure to include the correct unit in your answer.

Explanation:

Step1: Recall the formula for the area of a circle

The formula for the area of a circle is $A = \pi r^{2}$, where $r$ is the radius of the circle. Given $r = 8$ in, so $A=\pi\times8^{2}=64\pi$ in².

Step2: Determine the fraction of the circle represented by the sector

The central - angle of the sector is $\theta = 90^{\circ}$, and the total number of degrees in a circle is $360^{\circ}$. The fraction of the circle that the sector represents is $\frac{\theta}{360^{\circ}}=\frac{90^{\circ}}{360^{\circ}}=\frac{1}{4}$.

Step3: Calculate the area of the sector

The area of the sector $A_{s}$ is given by the fraction of the circle times the area of the circle. So $A_{s}=\frac{1}{4}\times A$. Substituting $A = 64\pi$ in², we get $A_{s}=\frac{1}{4}\times64\pi = 16\pi$ in².

Answer:

$16\pi$ in²