Question

a denoting the area of the sector of a circle of radius / formed by the central angle θ. find the missing quantity.
r = 6 inches; θ = 135° a =?
a = □ square inches
(type an integer or decimal rounded to three decimal places as needed.)

Explanation:

Step1: Convert the angle from degrees to radians

The formula for the area of a sector is \(A=\frac{1}{2}r^{2}\theta\), where \(\theta\) must be in radians.
We know that \(\theta\) (in radians) \(=\frac{\pi}{180}\times\theta\) (in degrees).
Given \(\theta = 135^{\circ}\), then \(\theta=\frac{\pi}{180}\times135=\frac{3\pi}{4}\) radians.

Step2: Substitute \(r = 6\) and \(\theta=\frac{3\pi}{4}\) into the area formula

\(A=\frac{1}{2}\times r^{2}\times\theta\)
Substitute \(r = 6\) and \(\theta=\frac{3\pi}{4}\) into the formula:
\(A=\frac{1}{2}\times6^{2}\times\frac{3\pi}{4}\)
First, calculate \(6^{2}=36\).
Then \(A=\frac{1}{2}\times36\times\frac{3\pi}{4}\)
\(A = 18\times\frac{3\pi}{4}=\frac{54\pi}{4}=\frac{27\pi}{2}\)

Step3: Calculate the numerical value

\(A=\frac{27\pi}{2}\approx\frac{27\times3.14159}{2}\)
\(A=\frac{84.823}{2}=42.4115\approx42.412\)

Answer:

\(42.412\)