Question

the measure of central angle rst is π radians. what is the area of the shaded sector? 4π units² 8π units² 16π units² 20π units²

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector with radius \( r \) and central angle \( \theta \) (in radians) is \( A=\frac{1}{2}r^{2}\theta \).

Step2: Identify the values of \( r \) and \( \theta \)

From the diagram, the radius \( r = 4 \) units and the central angle \( \theta=\pi \) radians.

Step3: Substitute the values into the formula

Substitute \( r = 4 \) and \( \theta=\pi \) into the formula \( A=\frac{1}{2}r^{2}\theta \).
First, calculate \( r^{2} \): \( r^{2}=4^{2} = 16 \).
Then, substitute into the formula: \( A=\frac{1}{2}\times16\times\pi \).
Simplify the expression: \( \frac{1}{2}\times16 = 8 \), so \( A = 8\pi \) square units.

Answer:

\( 8\pi \) units\(^{2}\) (corresponding to the option: 8π units²)