Question

I don't have a diagram, but here's the key info:

There is a sector of a circle:
The radius is 5.2
The arc length is 6π
You are not given the angle of the sector

Find the area of the sector, and give your answer in terms of pi I got 49π, let me know if I'm right ty. I've attached my working out

Answer 1

`(78)/(5) \pi = 15.6 \pi`

Explanation:

Given values:

• Radius (r) = 5.2
• Arc length = 6π

As the arc length is given in terms of pi, use the formulas where the angle is measured in radians.

`\boxed{\begin{minipage}{6.3 cm}\underline{Arc length}\\\\Arc length $=r \theta$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in radians.\\\end{minipage}}`

Substitute the given values into the arc length formula to calculate the central angle (in radians):

`\implies 6\pi=5.2\; \theta`

`\implies \theta=(6\pi)/(5.2)`

`\implies \theta=(15)/(13)\pi`

`\boxed{\begin{minipage}{6.3 cm}\underline{Area of a sector of a circle}\\\\Area of a sector = $\frac12 r^2 \theta$\\\\where:\\ \phantom{ww}$\bullet$ $r$ is the radius. \\ \phantom{ww}$\bullet$ $\theta$ is the angle measured in radians.\\\end{minipage}}`

Substitute the radius and the found angle into the formula and solve for area:

`\begin{aligned}\implies \textsf{Area of the sector}&=(1)/(2)(5.2)^2 \left((15)/(13)\pi\right)\\\\&=(1)/(2)(27.04) \left((15)/(13)\pi\right)\\\\&=(13.52) \left((15)/(13)\pi\right)\\\\&=(78)/(5)\pi \\\\&=15.6 \pi\end{aligned}`

Answer 2

• Area of sector is 15.6π

========================

Let the central angle be α.

Arc length equation

• s = 2πr × α/360, if α in degrees
• s = 2πr × α/2π = αr, if α in radians

We have

• s = 6π and r = 5.2

Find the central angle

• s = αr
• 6π = 5.2α
• α = 6π/5.2

Area of sector

• A = r²α/2, when α in radians

Substitute the value of α

• A = 5.2² × 6π/5.2 × 1/2 = 5.2 × 3π = 15.6π

Note: You could get it more easily if considered 360° = 2π in your calculations.