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 angle to radians

First, convert $\theta = 90^{\circ}$ to radians. We know that $180^{\circ}=\pi$ radians, so $90^{\circ}=\frac{\pi}{2}$ 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 $r = 4$ inches and $\theta=\frac{\pi}{2}$ radians. Substitute these values into the formula: $A=\frac{1}{2}\times4^{2}\times\frac{\pi}{2}$.

Step3: Calculate the area

First, calculate $4^{2}=16$. Then, $\frac{1}{2}\times16\times\frac{\pi}{2}=4\pi$.

Step4: Round the result

$4\pi\approx4\times3.14159 = 12.56636\approx12.566$ (rounded to three decimal places).

Answer:

$12.566$