Question

In the diagram below, arc AB has a central angle of 135° and an arclength of 15π. What is the area of the shaded sector?

Answer 1

I am aware of a way to do this, but it is rather difficult so, what is the easiest method to solve this equation?

Answer 2

in degrees, arc length = (theta/360) * 2 * pi * r

solve for the radius r. from there, the area of the sector is simply (135/360) * pi * r^2 , since you have 135/360 of a whole circle.

Answer 3

\[135^\circ=\frac{ 135 \pi }{180}=\frac{ 3\pi }{ 4 }~radians\]
\[\theta=\frac{ l }{ r }\]
\[\frac{ 3\pi }{ 4 }=\frac{ 15\pi }{ r }\]
\[r=15\pi \times \frac{ 4 }{ 3\pi} =20\]
\[area=\pi (20)^2\times \frac{ 135 }{ 360 }=?\]