Question

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

\( A=\frac{1\times25\times2\pi}{2\times36\times3} \).
Cancel out the common factor of \( 2 \) in the numerator and denominator: \( A = \frac{25\pi}{108} \).

Answer:

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