Question

find the area of this sector if ( r=\frac{2}{3} ) and ( \theta=\frac{pi}{6} ) radians. ( \frac{pi}{?} ) sq units

Explanation:

Step1: Recall the formula for the area of a sector

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

Step2: Substitute the given values into the formula

Given \(r = \frac{2}{3}\) and \(\theta=\frac{\pi}{6}\). Substitute these into the formula:
\(A=\frac{1}{2}\times(\frac{2}{3})^{2}\times\frac{\pi}{6}\)
First, calculate \((\frac{2}{3})^{2}=\frac{4}{9}\). Then the expression becomes \(A = \frac{1}{2}\times\frac{4}{9}\times\frac{\pi}{6}\)

Step3: Simplify the expression

\(\frac{1}{2}\times\frac{4}{9}=\frac{2}{9}\), and then \(\frac{2}{9}\times\frac{\pi}{6}=\frac{2\pi}{54}=\frac{\pi}{27}\)

Answer:

\(\frac{\pi}{27}\)