<< Chapter < Page Chapter >> Page >

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

215 π 18 = 37.525  units

Got questions? Get instant answers now!

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 .

Got questions? Get instant answers now!
Got questions? Get instant answers now!

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.

1.88

Got questions? Get instant answers now!

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.

Questions & Answers

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask