Question

find the area of the shaded region in the figure. (round your answer to two decimal places.) 120° 8 resources

Explanation:

Step1: Find area of sector

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $\theta = 120^{\circ}$ and $r = 8$.
$A_{sector}=\frac{120^{\circ}}{360^{\circ}}\times\pi\times8^{2}=\frac{1}{3}\times\pi\times64=\frac{64\pi}{3}$

Step2: Find area of triangle

The formula for the area of a triangle with two - side lengths $a = b=r = 8$ and included - angle $\theta=120^{\circ}$ is $A=\frac{1}{2}ab\sin\theta$.
$A_{triangle}=\frac{1}{2}\times8\times8\times\sin120^{\circ}=\frac{1}{2}\times64\times\frac{\sqrt{3}}{2}=16\sqrt{3}$

Step3: Find area of shaded region

The area of the shaded region $A_{shaded}=A_{sector}-A_{triangle}$.
$A_{shaded}=\frac{64\pi}{3}-16\sqrt{3}\approx\frac{64\times3.14159}{3}-16\times1.73205$
$=\frac{201.06176}{3}-27.7128\approx67.020587 - 27.7128\approx39.31$

Answer:

$39.31$