We can use our knowledge about the area of a circle to help us find the area of a sector. We know that the area of a circle is given by
\(A = \pi {r^2}\)
but if a sector is only a part of a circle, we can just find the area of the part. For example, since a full rotation of a circle is \(2\pi \) radians, we know that any smaller angle would be a fractional part of \(2\pi \). For example,
\(\pi radians \times \Large \frac{{1revolution}}{{2\pi radians}} = \Large \frac{\pi }{{2\pi }}revolutions = \Large \frac{1}{2}revolution\)
That is, the angle \(\pi \) radians is \(\frac{1}{2}\) of a revolution. Let’s generalize this:
\(\theta {\text{}}radians \times \Large \frac{{1{\text{}}revolution}}{{2\pi radians}} = \Large \frac{\theta }{{2\pi }}revolution\)
Then a sector whose angle measure is \(\theta \) is exactly \(\Large \frac{\theta }{{2\pi }}\) of a circle.
Then the area of a sector is \(\frac{\theta }{{2\pi }}\) times the area of a circle. That is,
\({A_{sector}} = \Large \frac{\theta }{{2\pi }} \times {A_{circle}}\)
\( = \Large \frac{\theta }{{2\pi }} \cdot \pi {r^2}\)
\( = \Large \frac{{\theta {r^2}}}{2}\)
Example: Find the area of the sector
Solution: We just need to substitute the angle and the radius into our formula. But first we note that
\(150^\circ \times \Large \frac{{\pi radians}}{{180^\circ }} = \Large \frac{{5\pi }}{6}radians\)
Then \(A = \Large \frac{\theta }{2}{r^2} = \Large \frac{1}{2}\left( {\Large \frac{{5\pi }}{6}} \right)\left( {{{10}^2}} \right) = \Large \frac{{500\pi }}{{12}} = \Large \frac{{125\pi }}{3}i{n^2}\)
Example: Find the area of the sector.
Solution: Again, we need to simply substitute our angle and radius into our formula. But we first need to convert \(240^\circ \) into radians. We have \(240^\circ \times \Large \frac{{\pi radians}}{{180^\circ }} = \Large \frac{{4\pi }}{3}radians\)
Then the area of the sector is
\(A = \Large \frac{\theta }{2}{r^2} = \Large \frac{1}{2} \cdot \Large \frac{{4\pi }}{3} \cdot {11^2} = \Large \frac{{484\pi }}{6} = \Large \frac{{242\pi }}{3}i{n^2}\)
Below you can download some free math worksheets and practice.