Question

find the area of a sector of a circle having radius r and central angle θ. r = 10.2 cm, θ = 63°. 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 $\theta = 63^{\circ}$ to radians. We know that $1^{\circ}=\frac{\pi}{180}$ radians. So $\theta = 63\times\frac{\pi}{180}=\frac{7\pi}{20}$ radians.

Step2: Use the 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 $r = 10.2$ cm and $\theta=\frac{7\pi}{20}$ radians. Substitute these values into the formula:
\[

$$\begin{align*} A&=\frac{1}{2}\times(10.2)^{2}\times\frac{7\pi}{20}\\ &=\frac{1}{2}\times104.04\times\frac{7\pi}{20}\\ & = 52.02\times\frac{7\pi}{20}\\ &=\frac{364.14\pi}{20}\\ & = 18.207\pi \end{align*}$$

\]

Step3: Calculate the numerical value

Now, calculate the value of $A$: $A = 18.207\pi\approx18.207\times3.14159\approx57.2$ $cm^{2}$

Answer:

$57.2$