Question

1. find the radius of the circle in the figure to the right. 2. find the area of a sector of a circle having radius r and central angle θ. r = 11.3 cm, θ = 81°. 3. the area is approximately □ cm². (do not round until the final answer. then round to the nearest tenth as needed.)

Explanation:

Step1: Convert degree to radian

The formula for the area of a sector is \(A=\frac{1}{2}r^{2}\theta\), where \(\theta\) is in radians.
We know that \(\theta = 81^{\circ}\), and to convert degrees to radians, we use the formula \(\theta_{rad}=\frac{\pi}{180}\times\theta_{deg}\).
So \(\theta=\frac{81\pi}{180}=\frac{9\pi}{20}\) radians.

Step2: Substitute values into the formula

Given \(r = 11.3\) cm.
Substitute \(r = 11.3\) and \(\theta=\frac{9\pi}{20}\) into \(A=\frac{1}{2}r^{2}\theta\).
\(A=\frac{1}{2}\times(11.3)^{2}\times\frac{9\pi}{20}\).
First, calculate \((11.3)^{2}=127.69\).
Then \(A=\frac{1}{2}\times127.69\times\frac{9\pi}{20}\).
\(A=\frac{127.69\times9\pi}{40}\).
\(A=\frac{1149.21\pi}{40}\).
Using \(\pi\approx3.1416\), we have \(A=\frac{1149.21\times3.1416}{40}\).
\(1149.21\times3.1416 = 1149.21\times(3 + 0.1416)=1149.21\times3+1149.21\times0.1416=3447.63+162.73=3610.36\).
\(A=\frac{3610.36}{40}=90.26\) (rounded to two decimal places).

Answer:

\(90.3\) \(cm^{2}\)