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Find the arc length along a circle of radius 10 units subtended by an angle of 215° .

215 π 18 = 37.525  units

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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. Recall that the area of a circle with radius r can be found using the formula A = π r 2 . If the two radii form an angle of θ , measured in radians, then θ 2 π 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 θ 2 π multiplied by the entire area. (Always remember that this formula only applies if θ is in radians.)

Area of sector = ( θ 2 π ) π r 2 = θ π r 2 2 π = 1 2 θ r 2

Area of a sector

The area of a sector    of a circle with radius r subtended by an angle θ , measured in radians, is

A = 1 2 θ r 2

See [link] .

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 slice is labeled: A equals one half times theta times r squared.
The area of the sector equals half the square of the radius times the central angle measured in radians.

Given a circle of radius r , find the area of a sector defined by a given angle θ .

  1. If necessary, convert θ to radians.
  2. Multiply half the radian measure of θ by the square of the radius r : A = 1 2 θ r 2 .

Finding the area of a sector

An automatic lawn sprinkler sprays a distance of 20 feet while rotating 30 degrees, as shown in [link] . 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.
The sprinkler sprays 20 ft within an arc of 30° .

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:

30  degrees = 30 π 180 = π 6  radians

The area of the sector is then

Area = 1 2 ( π 6 ) ( 20 ) 2 104.72

So the area is about 104.72  ft 2 .

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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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Use linear and angular speed to describe motion on a circular path

In addition to finding the area of a sector, we can use angles to describe the speed of a moving object. An object traveling in a circular path has two types of speed. Linear speed is speed along a straight path and can be determined by the distance it moves along (its displacement ) in a given time interval. For instance, if a wheel with radius 5 inches rotates once a second, a point on the edge of the wheel moves a distance equal to the circumference, or 10 π inches, every second. So the linear speed of the point is 10 π in./s. The equation for linear speed is as follows where v is linear speed, s is displacement, and t is time.

v = s t

Angular speed results from circular motion and can be determined by the angle through which a point rotates in a given time interval. In other words, angular speed    is angular rotation per unit time. So, for instance, if a gear makes a full rotation every 4 seconds, we can calculate its angular speed as 360  degrees 4  seconds = 90 degrees per second. Angular speed can be given in radians per second, rotations per minute, or degrees per hour for example. The equation for angular speed is as follows, where ω (read as omega) is angular speed, θ is the angle traversed, and t is time.

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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