Arcs and Sectors: Learn It 2

Finding the Area of a Sector of a Circle

In addition to arc length, we can also use angles to find the area of a sector of a circle. A sector is a region of a circle bounded by two radii and the intercepted arc, like a slice of pizza or pie.
The area of a circle with radius [latex]r[/latex] can be found using the formula [latex]A=\pi {r}^{2}[/latex].

If the two radii form an angle of [latex]\theta[/latex], measured in radians, then [latex]\frac{\theta }{2\pi }[/latex] is the ratio of the angle measure to the measure of a full rotation and is also, therefore, the ratio of the area of the sector to the area of the circle. Thus, the area of a sector is the fraction [latex]\frac{\theta }{2\pi }[/latex] multiplied by the entire area. (Always remember that this formula only applies if [latex]\theta[/latex] is in radians.)

[latex]\begin{align}\text{Area of sector}&=\left(\frac{\theta }{2\pi }\right)\pi {r}^{2} \\ &=\frac{\theta \pi {r}^{2}}{2\pi } \\ &=\frac{1}{2}\theta {r}^{2} \end{align}[/latex]

area of a sector

The area of a sector of a circle with radius [latex]r[/latex] subtended by an angle [latex]\theta[/latex], measured in radians, is

[latex]A=\frac{1}{2}\theta {r}^{2}[/latex]

Graph showing a circle with angle theta and radius r, and the area of the slice of circle created by the initial side and terminal side of the angle.
The area of the sector equals half the square of the radius times the central angle measured in radians.
How To: Given a circle of radius [latex]r[/latex], find the area of a sector defined by a given angle [latex]\theta[/latex].

  1. If necessary, convert [latex]\theta[/latex] to radians.
  2. Multiply half the radian measure of [latex]\theta[/latex] by the square of the radius [latex]r:\text{ } A=\frac{1}{2}\theta {r}^{2}[/latex].
An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees. What is the area of the sector of grass the sprinkler waters?

Illustration of a 30 degree ange with a terminal and initial side with length of 20 feet.
The sprinkler sprays 20 ft within an arc of 30°.

In central pivot irrigation, a large irrigation pipe on wheels rotates around a center point. A farmer has a central pivot system with a radius of 400 meters. If water restrictions only allow her to water 150 thousand square meters a day, what angle should she set the system to cover? Write the answer in radian measure to two decimal places.