Question

find the area of this sector if ( r = 4 ) and ( \theta=\frac{pi}{12} ) radians.
( \frac{? pi}{square} ) 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 \(r = 4\) and \(\theta=\frac{\pi}{12}\), we substitute these into the formula:
\(A=\frac{1}{2}\times(4)^{2}\times\frac{\pi}{12}\)
First, calculate \((4)^{2}=16\). Then the expression becomes \(\frac{1}{2}\times16\times\frac{\pi}{12}\).
\(\frac{1}{2}\times16 = 8\). So now we have \(8\times\frac{\pi}{12}\).
Simplify \(\frac{8\pi}{12}\) by dividing numerator and denominator by 4. \(\frac{8\pi\div4}{12\div4}=\frac{2\pi}{3}\)

Answer:

\(\frac{2\pi}{3}\)