Question

find the area of a sector of a circle having radius r and central angle θ. r = 12.1 cm, θ = 85°. the area is approximately cm². (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Convert angle to radians

First, convert $\theta = 85^{\circ}$ to radians. Use the conversion factor $\theta_{rad}=\theta_{deg}\times\frac{\pi}{180}$. So $\theta = 85\times\frac{\pi}{180}=\frac{17\pi}{36}$ radians.

Step2: Use sector - area formula

The formula for the area of a sector of a circle is $A=\frac{1}{2}r^{2}\theta$, where $r$ is the radius and $\theta$ is the central - angle in radians. Given $r = 12.1$ cm and $\theta=\frac{17\pi}{36}$ radians. Substitute these values into the formula: $A=\frac{1}{2}\times(12.1)^{2}\times\frac{17\pi}{36}$.

Step3: Calculate the area

First, $(12.1)^{2}=146.41$. Then $\frac{1}{2}\times146.41\times\frac{17\pi}{36}=\frac{146.41\times17\pi}{72}$.
$146.41\times17 = 2488.97$. So $A=\frac{2488.97\pi}{72}\approx\frac{2488.97\times3.14159}{72}$.
$2488.97\times3.14159 = 7819.777$. Then $A=\frac{7819.777}{72}\approx108.6$.

Answer:

$108.6$