Question

a denotes the area of the sector of a circle of radius r formed by the central angle θ. find the missing quantity. r = 4 inches, θ = 90°, 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

We know that \(180^{\circ}=\pi\) radians. So, for \(\theta = 90^{\circ}\), \(\theta=\frac{90^{\circ}}{180^{\circ}}\times\pi=\frac{\pi}{2}\) radians.

Step2: Use the formula for the area of a sector

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 \(r = 4\) inches and \(\theta=\frac{\pi}{2}\) radians.
Substitute the values into the formula: \(A=\frac{1}{2}\times(4)^{2}\times\frac{\pi}{2}\).
First, calculate \((4)^{2}=16\). Then \(\frac{1}{2}\times16 = 8\).
So, \(A = 8\times\frac{\pi}{2}=4\pi\).
Using \(\pi\approx3.14159\), we get \(A\approx4\times3.14159 = 12.56636\approx12.566\)

Answer:

\(12.566\)