# 7.1 Angles  (Page 7/29)

 Page 7 / 29

Find the arc length along a circle of radius 10 units subtended by an angle of $\text{\hspace{0.17em}}215°.$

## 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 $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ can be found using the formula $\text{\hspace{0.17em}}A=\pi {r}^{2}.\text{\hspace{0.17em}}$ If the two radii form an angle of $\text{\hspace{0.17em}}\theta ,$ measured in radians, then $\text{\hspace{0.17em}}\frac{\theta }{2\pi }\text{\hspace{0.17em}}$ 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 $\text{\hspace{0.17em}}\frac{\theta }{2\pi }\text{\hspace{0.17em}}$ multiplied by the entire area. (Always remember that this formula only applies if $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is in radians.)

## Area of a sector

The area of a sector    of a circle with radius $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ subtended by an angle $\text{\hspace{0.17em}}\theta ,$ measured in radians, is

$A=\frac{1}{2}\theta {r}^{2}$

Given a circle of radius $\text{\hspace{0.17em}}r,$ find the area of a sector defined by a given angle $\text{\hspace{0.17em}}\theta .$

1. If necessary, convert $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ to radians.
2. Multiply half the radian measure of $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ by the square of the radius $\text{\hspace{0.17em}}r:\text{​}A=\frac{1}{2}\theta {r}^{2}.$

## 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?

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:

The area of the sector is then

$\begin{array}{ccc}\hfill \text{Area}& =& \frac{1}{2}\left(\frac{\pi }{6}\right){\left(20\right)}^{2}\hfill \\ & \approx & 104.72\hfill \end{array}$

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

## 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 $\text{\hspace{0.17em}}10\pi \text{\hspace{0.17em}}$ inches, every second. So the linear speed of the point is $\text{\hspace{0.17em}}10\pi \text{\hspace{0.17em}}$ in./s. The equation for linear speed is as follows where $\text{\hspace{0.17em}}v\text{\hspace{0.17em}}$ is linear speed, $\text{\hspace{0.17em}}s\text{\hspace{0.17em}}$ is displacement, and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is time.

$v=\frac{s}{t}$

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 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 $\text{\hspace{0.17em}}\omega \text{\hspace{0.17em}}$ (read as omega) is angular speed, $\text{\hspace{0.17em}}\theta \text{\hspace{0.17em}}$ is the angle traversed, and $\text{\hspace{0.17em}}t\text{\hspace{0.17em}}$ is time.

x exposant 4 + 4 x exposant 3 + 8 exposant 2 + 4 x + 1 = 0
x exposent4+4x exposent3+8x exposent2+4x+1=0
HERVE
How can I solve for a domain and a codomains in a given function?
ranges
EDWIN
Thank you I mean range sir.
Oliver
proof for set theory
don't you know?
Inkoom
find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
factoring polynomial
find general solution of the Tanx=-1/root3,secx=2/root3
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
where can I get indices
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Need help with this question please
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
The sequence is {1,-1,1-1.....} has