### 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

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

$$A = \Large \frac{\theta }{2}{r^2} = \Large \frac{1}{2} \cdot \Large \frac{{4\pi }}{3} \cdot {11^2} = \Large \frac{{484\pi }}{6} = \Large \frac{{242\pi }}{3}i{n^2}$$

7528 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:

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5107 x

Find the area of each sector.

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

Example of one question:

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3937 x

Find the area of each sector.

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

Example of one question:

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### Geometry

Circles
Congruent Triangles
Constructions
Parallel Lines and the Coordinate Plane
Properties of Triangles

### Algebra and Pre-Algebra

Beginning Algebra
Beginning Trigonometry
Equations
Exponents
Factoring
Linear Equations and Inequalities
Percents
Polynomials