Radian Measure, Arc Length, and Angular and Linear Speed

1. Radian Measure and Its Relationship to Degree Measure

1a. The Definition of a Radian

Although we are used to measuring angles with degrees, with representing a full rotation around a circle, and as a result representing of one full rotation, this choice is arbitrary and is not related to anything tangible. In other words, by this standard, angles could be measured to represent any fraction of a full rotation (hundredths, twentieths, twelfths, etc.).

Consider a circle whose radius is r, centered at the origin.

The angle with vertex at center is selected so that the length of the circular arc intercepted by the angle is equal to the radius of the circle, as shown in the figure.

Since the angle has its vertex at the center of a circle, it is called a central angle. The measure of the angle in the figure is defined as 1 radian.

Now, consider an arc that has length s. Since each radian corresponds to a circular arc of length r, the quantity gives the measure of the central angle, measured in radians.

formula to know

Radian Measure of a Central Angle

where s is the length of the arc and r is the radius of the circle that contains the arc.

Since is the ratio of two lengths, its measure is unitless. It is not necessary (but is sometimes helpful) to write the label “radians” after an angle measure. If there is no degree symbol affixed to the angle measure, it is assumed that the angle is measured in radians.

big idea

Because an angle defined without a symbol is assumed to be measured in radians, it is very important to use the degree symbol when degrees are intended.

terms to know

Central Angle

An angle measured from the center of a circle which is used to define the arc of a circle.

Radian

The measure of the central angle that corresponds to the circular arc whose length is the same as the radius of the circle that contains the arc.

1b. Converting Between Radian and Degree Measure

Now, let’s try to establish a relationship between degree measure and radian measure. Recall that the circumference of a circle is where r is the radius of the circle.

Consider the entire circumference of the circle, which is What angle in radians corresponds to a full rotation, meaning that the length of the circular arc is the circumference of the circle?

Since gives the central angle corresponding to a circular arc with length s, let

Then, (since the radius is naturally not equal to 0).

Thus, the central angle represents one full circular rotation. Recall that the angle also represents one full circular rotation.

This means that These angles are the same, but they are represented two different ways.

If we divide both sides of the equation by 2, we have This equation is often easier to remember.

Now, we’ll use the equation to establish formulas for 1 radian in terms of degrees and in terms of radians.

• Divide both sides of the equation by 180:
• Divide both sides of the equation by

big idea

To convert an angle from degree measure to radian measure, multiply by

To convert an angle from radian measure to degree measure, multiply by

EXAMPLE

Convert the angle to radian measure, leaving the result in terms of

To convert, multiply by

	Multiply 45 by
	Perform the operation.
	Reduce the fraction.

Thus, is equivalent to radian.

We will see throughout this course that some angles in radians are left as multiples of while some are approximated.

try it

Consider the angle

Convert the angle to radian measure. Give your answer in exact form.

To convert from degrees to radians, multiply by and simplify:

radians

Now, we’ll convert from radian measure to degree measure.

EXAMPLE

Convert 1.2 radians to degree measure. Round to the nearest tenth of a degree.

To convert, multiply 1.2 by

	Multiply 1.2 by
	Use a calculator to approximate.

Thus, 1.2 radians is approximately

try it

Consider the angle measured in radians.

Convert the angle to degree measure, rounded to the nearest tenth.

To convert from radians to degrees, multiply by then round as stated in the directions.

2. Finding the Length of a Circular Arc

Using the formula we can multiply both sides by r to obtain a formula that gives the length of the arc when the radius of the circle and the angle, measured in radians, are known.

formula to know

Length of a Circular Arc

where r is the radius of the circle and is the measure of the central angle, in radians.

EXAMPLE

The circular arc below has a central angle measuring What is the exact length of the arc?



To use the formula the angle must be measured in radians.

First, convert to radians.

	Multiply the angle by
	Multiply and reduce.

Then, use in the formula to find the length.

	This is the formula for the length of the arc.
	Substitute and
	Simplify.

The length of the circular arc is inches.

try it

A circular arc has radius 16 cm and has central angle

What is the exact length of the circular arc?

To find the length of the arc, use the formula Be very careful though, since needs to be measured using radians.

First, convert to radians:

Now, find the length of the arc:

3. Solving Problems Involving Angular and Linear Speed

3a. Angular Speed

Have you ever played a CD and watched it spin and noticed how fast it goes? Depending on where the laser is reading the CD, it could spin anywhere between 200 and 500 revolutions per minute (rpm). This type of speed is an example of angular speed, a measure of how a rotation angle changes over time. The “revolutions” in rpm indicate that there is motion around a circle, as opposed to a straight line.

formula to know

Angular Speed

where is the angle of rotation and t is the time elapsed.
Note that is the Greek letter omega.

Let’s start by solving some problems using angular speed.

EXAMPLE

A water wheel completes one full rotation every 5 seconds. What is its angular speed, measured in radians per second? Round your final answer to three decimal places.

Since the wheel makes one full rotation,

Since the time elapsed is 5 seconds, we have

Approximating to three decimal places, the angular speed is approximately 1.257 radians per second.

Sometimes we are given an angular speed in terms of revolutions per minute. Remember that each revolution means the angle rotated radians.

EXAMPLE

An old vinyl record turns at a rate of 45 revolutions per minute. What is its angular speed in radians per second?

There could be several ways to address the difference in time units.

Since the record is turning at a rate of 45 revolutions per minute, we can change this to revolutions per second as follows:

	Write the angular speed as a ratio.
	Since each minute has 60 seconds, multiply by a conversion factor.
	Since there are minutes in the numerator and denominator, they cancel.
	Simplify.

Thus, the record makes of a revolution every second. Since each revolution is radians, the angular speed is

try it

The wheel of a bicycle completes 100 revolutions every minute.

What is the angular speed of the wheel in radians per second? Round your answer to the nearest hundredth.

The wheel completes 100 revolutions per minute. Since each revolution is radians, the wheel travels radians each minute.

Now, we perform a unit conversion to get the desired units, radians per second:

Thus, the angular speed of the wheel is 10.47 radians per second.

term to know

Angular Speed

The change in the angle of rotation per unit of time.

3b. Linear Speed

When you are traveling in a car at a rate of 60 miles per hour, this is an example of linear speed.

When traveling around a circle, the total distance traveled is the length of the circular arc, s. Since the distance traveled is affected by the angle of rotation as well as the radius of the circular path.

formula to know

Linear Speed

where s is the length of the arc (distance traveled) and t is the time interval.
Since assuming that linear and angular speed are measured using the same units of time.

EXAMPLE

A CD has a diameter of 12 centimeters. When playing audio, the angular speed varies to keep the linear speed constant where the disc is being read. When reading along the outer edge of the disc, the angular speed is about 400 revolutions per minute. What is its linear speed?

When reading the outer edge of the CD, the diameter of the CD is 12 cm, which means the radius is 6 cm.

	The CD is spinning at 400 revolutions per minute, and each revolution has a rotation angle of
	The rotation is happening at a rate of 400 revolutions per minute, which is radians per minute. This is the angular speed in radians per minute.
	Use the formula

The linear speed of the CD is

watch

In this video, we’ll find the angular speed of a bicycle with 24-inch diameter tires that is traveling at 15 miles per hour.

try it

A truck with 32-inch diameter wheels is traveling at 60 miles per hour.

What is the angular speed of the wheels in radians per minute?

The radius of the truck is inches. The linear speed is 60 miles per hour, which means the length of the arc is 60 miles. To find we first need to make the distances have the same units:

Then, using the equation we have or radians.

This is the number of radians in one hour, which means we need to now convert this to radians per minute:

Thus, the angular speed is 3,960 radians per minute.

How many revolutions per minute do the wheels make? Round to the nearest whole number.

There are radians in every revolution. We convert as follows:

Thus, the wheels make about 630 revolutions per minute.

term to know

Linear Speed

The change in distance that an object travels per unit of time.

summary

In this lesson, you learned about another unit of measurement of an angle, known as a radian. A radian measure of an angle is defined as the ratio of the length of a circular arc to the length of the circle's radius. You also explored radian measure and its relationship to degree measure, using the equation to establish formulas for 1 radian in terms of degrees and in terms of radians, which then allowed you to convert between radian and degree measure. You also learned how to use radians to find the length of a circular arc and to solve problems with angular speed (a measure of how a rotation angle changes over time) and linear speed (the change in distance traveled per unit of time) for an object that is moving around a circular path.

SOURCE: THIS TUTORIAL HAS BEEN ADAPTED FROM OPENSTAX "PRECALCULUS” BY JAY ABRAMSON. ACCESS FOR FREE AT OPENSTAX.ORG/DETAILS/BOOKS/PRECALCULUS-2E. LICENSE: CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL.