Question

find the area of a sector of a circle with radius 6 centimeters and central angle \\( \frac { 4 \pi } { 3 } \\) radians. if you leave your answer in terms of \\( \pi \\), write out the word pi. if you leave your answer as a decimal, round to the hundredths place. square centimeters

Explanation:

Step1: Recall 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 of the circle and \(\theta\) is the central angle in radians.

Step2: Substitute the given values into the formula

Given \(r = 6\) cm and \(\theta=\frac{4\pi}{3}\) radians.
Substitute into the formula: \(A=\frac{1}{2}(6)^{2}\times\frac{4\pi}{3}\)
First, calculate \((6)^{2}=36\). Then the expression becomes \(\frac{1}{2}\times36\times\frac{4\pi}{3}\)
\(\frac{1}{2}\times36 = 18\), and \(18\times\frac{4\pi}{3}\)
\(18\div3 = 6\), \(6\times4\pi=24\pi\)

If we want a decimal approximation, since \(\pi\approx3.14\), then \(A = 24\times3.14=75.36\)

Answer:

\(24\pi\) (or \(75.36\)) square centimeters