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Arc length and sector area

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

Circles-Area-of-a-Sector-1

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.

Circles-Area-of-a-Sector-2

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.


Downloads:
4583 x

Find the length of each arc. Round your answers to the nearest tenth.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Arc-length-and-sector-area-Easy

Watch below how to solve this example:

 

Downloads:
2829 x

Find the area of each sector.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Arc-length-and-sector-area-Medium

Watch below how to solve this example:

 

Downloads:
2214 x

Find the area of each sector.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Circles-Arc-length-and-sector-area-Hard

Watch below how to solve this example:

 
 
 

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