Example 9 from Algebra and Trigonometry, 7.1 Angles

Finding the Area of a Sector

An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in Figure 23. What is the area of the sector of grass the sprinkler waters?

Illustration of a 30-degree angle with a terminal and initial side with length of 20 feet.

Figure 23 The sprinkler sprays 20 ft within an arc of

30° .

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

Solution
First, we need to convert the angle measure into radians. Because 30 degrees is one of our special angles, we already know the equivalent radian measure, but we can also convert:

$\begin{matrix} 30 degrees & = & 30 \cdot \frac{π}{180} \\ = & \frac{π}{6} radians \end{matrix}$

The area of the sector is then

$\begin{matrix} Area & = & \frac{1}{2} (\frac{π}{6}) {(20)}^{2} \\ \approx & 104.72 \end{matrix}$
So the area is about $104.72 {ft}^{2} .$

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

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.

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Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Solution

Your turn

Solution

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