Question

find the area of this sector if ( r = \frac{5}{6} ) and ( \theta = \frac{2pi}{3} ) radians.

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 \(r = \frac{5}{6}\) and \(\theta=\frac{2\pi}{3}\).
First, calculate \(r^{2}\): \(r^{2}=(\frac{5}{6})^{2}=\frac{25}{36}\).
Then, substitute into the formula: \(A=\frac{1}{2}\times\frac{25}{36}\times\frac{2\pi}{3}\).

Step3: Simplify the expression

\(\frac{1}{2}\times\frac{25}{36}\times\frac{2\pi}{3}=\frac{25\pi}{108}\)

Answer:

\(\frac{25\pi}{108}\)