Question

A central angle of 2 radians off an arc of length 6 inches. find the area of the sector formed

Answer 1

`\textit{arc's length}\\\\ s = r\theta ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ \theta =2\\ s=6 \end{cases}\implies 6=r2\implies \cfrac{6}{2}=r\implies 3=r \\\\[-0.35em] ~\dotfill\\\\ \textit{area of a sector of a circle}\\\\ A=\cfrac{\theta r^2}{2} ~~ \begin{cases} r=radius\\ \theta =\stackrel{radians}{angle}\\[-0.5em] \hrulefill\\ r=3\\ \theta =2 \end{cases}\implies A=\cfrac{(2)(3)^2}{2}\implies A=9~in^2`

Answer 2

The formula for the area of a sector is (θ/2) * r^2, where θ is the central angle in radians and r is the radius of the circle.

Since the central angle is 2 radians and the arc length is 6 inches, we can use the formula s = rθ to find the radius:

6 inches = r * 2 radians
r = 3 inches

Now we can use the formula for the area of a sector:

Area = (2/2) * 3^2 * π
Area = 9π square inches

Therefore, the area of the sector is 9π square inches.