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6. Radian Measure and Arc Length
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Chapter 5
6. 

Radian Measure and Arc Length

In this comprehensive lesson, you will explore the concept of radian measure and its application in calculating the length of the arc of a circle. Radians are an alternative unit for measuring angles, and they offer several advantages over degrees. By understanding the definition of a radian as the measure of a central angle intercepting an arc equal to the radius, you will grasp the fundamental concept of radian measure. This lesson covers the conversion between degrees and radians, allowing you to work seamlessly with both units. You will discover conversion formulas that enable you to convert angle measures from degrees to radians and vice versa. These conversion techniques will be instrumental in applying radian measure to real-world problems. By exploring the relationship between arc length and the radius of a circle, you will derive the formula for finding the length of an arc using radian measure. Real-life scenarios, including clock hands and inscribed angles, will be used to demonstrate the practicality of radian measure in determining arc lengths. To solidify your understanding, you will tackle a challenging problem involving a satellite orbiting the Moon. By calculating the distance traveled by the satellite when it completes 60% of its orbit, you will witness the direct application of radian measure in a real astronomical context. By the end of this lesson, you will have a comprehensive understanding of radian measure, its benefits, and its practical use in calculating arc lengths. You will gain the skills to convert between degrees and radians and apply radian measure to various scenarios involving circles and angles.
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Radian Measure and Arc Length
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Angles can be measured not only by degrees but also by radians. In this lesson, the concept of radian measure will be introduced. Additionally, the formulas related to arc length will be derived and exercised.

Catch-Up and Review

Here are a few recommended readings before getting started with this lesson.

Try your knowledge on these topics.

a Calculate the circumference of the given circle.
A circle with radius 4

Round the answer to the closest integer.

b Find the measure of arc AB.
A circle with an inscribed angle that measures 35 degrees and intercepts arc AB
c Determine the measures of ∠ SOR and ∠ STR.
A circle with an arc that measures 105 degrees intercepted by a central angle SOR and inscribed angle STR
Challenge

Investigating the Orbit of a Satellite

In 2007, a satellite named SELENE was launched to explore the Moon's surface for a few years. It orbited the Moon in a circular path staying at a constant altitude of 100 kilometers. Given that the radius of the Moon is 1737 kilometers, find the distance that the satellite traveled when it completed 60 % of its orbit.

The Moon and a satellite
External credits: @wirestock
Discussion

Radian

In order to introduce the concept of radian measure, the definition of a radian should first be explored.

A radian, like a degree, is an angle unit. One radian is defined as the measure of the central angle that intercepts an arc equal in length to the radius of the circle. It corresponds to roughly 57.3^(∘).

Animation of forming an arc that measures 1 radian
If the arc length is 2 radii, the measure of the corresponding central angle is 2 radians, and so on. Therefore, radians describe the number of radii an angle creates on a circle.
Rotation of a point on a circle
It can be observed that a semicircle corresponds to an arc length of π radii and the circumference of a circle corresponds to an arc length of 2π radii.
Rotation of a circle along a segment
In calculations, even if the angle is given in radians, rad is seldom written. Instead, no unit marker indicates radians. Consider two expressions. cos 64^(∘) and cos 5

The first angle is given in degrees, and the other is given in radians. At first glance, radians might seem inconvenient, but they make calculations simpler in certain circumstances,. Radians are also the SI unit for angles.
Rule

Relationship Between Radians and Degrees

Radians and degrees are two different units of measure for an angle. Knowing how to convert between the two is majestic, especially in trigonometry. Recall that a full circle measures 360^(∘), which corresponds to 2π radians. 360^(∘) = 2 π rad ⇔ 180^(∘) = π rad Using this equality, it is possible to find equivalent expressions for 1^(∘) and 1rad.

Degrees to Radians Radians to Degrees
1^(∘) = π/180rad 1rad = 180^(∘)/π

Proof
The arc length of one revolution around a circle is the circumference of the circle 2π r. The number of radians in a full revolution is calculated by dividing 2π r by the radius r of the circle. 2π r/r = 2π This means that one revolution is 2π radians which in degrees is 360^(∘), and that leads to the expression 360^(∘) = 2π rad. Both sides then divided by 2 gives the following equation. 180^(∘) = π rad Therefore, 180^(∘) correspond to π radians.

From the relation 180^(∘) = π rad, it is possible to find two rules by dividing both sides by either 180 or π. To find an expression for 1^(∘), divide both sides by 180.

180 ^(∘) = π rad
DivEqn

.LHS /180.=.RHS /180.

180/180 ^(∘) = π/180 rad
CalcQuot

Calculate quotient

1 ^(∘) = π/180 rad

As shown, 1^(∘) corresponds to π180rad, which is approximately 0.017 radians. To get an expression for 1rad, divide both sides by π.

180 ^(∘) = π rad
DivEqn

.LHS /π.=.RHS /π.

180^(∘)/π = π/π rad
CalcQuot

Calculate quotient

180^(∘)/π = 1 rad
RearrangeEqn

Rearrange equation

1 rad = 180^(∘)/π

Therefore, 1 radian corresponds to 180^(∘)π, which is approximately 57.3^(∘).

A measure expressed in degrees can be converted into radians by multiplying by π180. Similarly, multiplying a measure expressed in radians by 180^(∘)π will convert it into degrees.
Pop Quiz

Practice Converting Degrees Into Radians

Convert degrees into radians. Round your answer to the second decimal place. Do not include the unit abbreviation rad in the answer.

Convert into radians
Pop Quiz

Practice Converting Radians Into Degrees

Convert radians into degrees. Round your answer to the closest integer, and do not include the degree symbol in the answer.

Convert into degrees
Explore

Investigating the Radian Measure for Different Circles

Consider a circle with a radius of 3 units. Move the point on the circle and pay close attention to the ratio of the arc length to the radius as the arc measure changes.

Next, consider a circle with an arc whose measure is constant. Examine the ratio of the arc length to the radius as the radius changes.

In the second case, why is the ratio constant? What does this say about the relationship between the arc length and the radius?
Discussion

Comparing Radian Measures of Concentric Circles

To understand the observed relation, consider two concentric circles with different radii r_1 and r_2.

Two concentric circles with different radii, the same central angle and two intercepted arcs on each circle

These two circles are similar, as all circles are similar under dilation. Therefore, the ratios of corresponding parts of the circles are proportional. This means the ratio of the radii is equal to the ratio of the arc lengths intercepted by the same central angle. r_1/r_2=s_1/s_2 This proportion can be rewritten as an equivalent equation.

r_1/r_2=s_1/s_2
▼
Rearrange equation
MultEqn

LHS * r_2s_2=RHS* r_2s_2

r_1/r_2* r_2s_2=s_1/s_2 * r_2s_2
CancelCommonFac

Cancel out common factors

r_1/r_2* r_2s_2=s_1/s_2 * r_2s_2
SimpQuotProd

Simplify quotient and product

r_1s_2=s_1r_2
DivEqn

.LHS /r_1.=.RHS /r_1.

s_2=s_1r_2/r_1
MovePartNumRight

a* b/c=a/c* b

s_2=s_1/r_1r_2
DivEqn

.LHS /r_2.=.RHS /r_2.

s_2/r_2=s_1/r_1

The obtained proportion shows that the ratio of the length of an arc intercepted by a certain central angle to the radius of the circle is constant in every circle. In other words, the arc length is proportional to the radius. This fact leads to the definition of radian measure.
Discussion

Using Radians to Calculate Arc Lengths

The radian measure of a central angle is defined as the quotient between the length of the arc intercepted by the angle and the radius of the circle.

A circle with origin O, radius r, central angle θ, and the intercepted arc by the angle, denoted as s.

Based on the diagram above, the radian measure of the central angle ∠ O is defined as follows.

θ=s/r

The arc length is proportional to the radius, so the radian measure θ of the central angle is a constant of proportionality. The radian measure is expressed in radians.

By using this definition, the formula for the length of an arc can be derived.

s=θ r

Here, θ is the angle measure given in radians. However, what if the measure θ of a central angle is given in degrees? In that case, by multiplying θ by the conversion factor π180^(∘), the measure can be converted into radians and substituted into the formula. s=( π/180^(∘)θ) r [0.3cm] ⇕ s=π r(θ/180^(∘)) This formula is often written in the following equivalent manner.

s=2π r(θ/360^(∘))

This equivalent form is convenient to work with because 2π r is the circumference of a circle. Since a full circle measures 360^(∘), dividing 2π r by 360 results in the length of an arc intercepted by a 1-degree central angle. Finally, by multiplying that value by the measure of the arc, its length is obtained.

Animation of obtaining arc length formula
Example

Arc Length Between the Long and Short hand of a Clock

In her room Paulina has a big clock whose longer hand is 9 centimeters long. When Paulina came home and looked at her clock, it was exactly 2P.M. She calculated the angle between the longer and shorter hands of the clock to be 60^(∘).

After a Geometry lesson, Paulina started wondering how long the arc formed by the hands of the clock is. Based on the length of the longer hand, she estimates the radius of the clock to be 15 centimeters. Help Paulina calculate the length of the arc. Round the answer to one decimal place.

Hint
Since the central angle is given in degrees, the arc length can be calculated using the formula s=2π r( θ360^(∘)).

Solution
To find the length of an arc using the measure of the corresponding central angle given in degrees, the following formula can be used. s=2π r (θ/360^(∘)) It is known that the radius of the clock is about 15 centimeters and the central angle that intercepts the arc measures 60^(∘).

By substituting these values into the above formula, the length of the arc can be determined.

s=2π r (θ/360^(∘))
SubstituteII

r= 15, θ= 60^(∘)

s=2π ( 15) (60^(∘)/360^(∘))
▼
Evaluate right-hand side
ReduceFrac

a/b=.a /60^(∘)./.b /60^(∘).

s=2π (15) (1/6)
MoveLeftFacToNumOne

a* 1/b= a/b

s=2π(15)/6
Multiply

Multiply

s=30π/3
ReduceFrac

a/b=.a /3./.b /3.

s=10π
UseCalc

Use a calculator

s=31.415926...
RoundDec

Round to 1 decimal place(s)

s≈ 31.4

The arc between the hands of the clock is about 31.4 centimeters long.

Example

Finding the Length of an Arc

After having dinner, Paulina decided to do her math homework. She is given a circle with radius 5 inches and an inscribed angle that measures 40^(∘).

A circle with a inscribed angle MKN that measures 40 degrees

Paulina is asked to convert the measure of the given angle into radians and then use it to find the length of MN. Help Paulina find the correct answer. The length should be rounded to the closest integer.

Hint
To convert the measure of an angle from degrees to radians, use the conversion factor π180^(∘). The length of the arc can be found by using the formula s=θ r, where θ is the measure of the corresponding central angle in radians.

Solution
It can be seen in the diagram that ∠ MKN, which is an inscribed angle, measures 40^(∘). By multiplying this value by the conversion factor π180^(∘), the measure of this angle can be converted from degrees to radians.

m∠ MKN = 40^(∘) * π/180^(∘)
MoveLeftFacToNum

a*b/c= a* b/c

m∠ MKN = 40^(∘) π/180^(∘)
ReduceFrac

a/b=.a /20^(∘)./.b /20^(∘).

m∠ MKN = 2π/9

Next, to find the length of MN, the measure of the corresponding central angle should be known. Recall that the measure of the inscribed angle is half the measure of the corresponding central angle. In this case, ∠ MKN corresponds to a central angle ∠ MON.

A circle with inscribed angle MKN and corresponsing central angle MON

Using this information, the measure of ∠ MON can be found.

m∠ MKN=1/2m∠ MON
Substitute

m∠ MKN= 2π/9

2π/9=1/2m∠ MON
▼
Solve for m∠ MON
MultEqn

LHS * 2=RHS* 2

2π/9(2)=m∠ MON
MoveRightFacToNum

a/c* b = a* b/c

2π(2)/9=m∠ MON
Multiply

Multiply

4π/9=m∠ MON
RearrangeEqn

Rearrange equation

m∠ MON=4π/9

Finally, the length of MN can be calculated using the corresponding formula. s=θ r ⇓ MN=m∠ MON r Here, 4π9 and 5 can be substituted for m∠ MON and r, respectively.

MN=m∠ MON r
SubstituteII

m∠ MON= 4π/9, r= 5

MN=( 4π/9) ( 5)
▼
Evaluate right-hand side
MoveRightFacToNum

a/c* b = a* b/c

MN=4π(5)/9
Multiply

Multiply

MN=20π/9
UseCalc

Use a calculator

MN=6.981317...
RoundInt

Round to nearest integer

MN≈ 7

The length of MN is approximately 7 inches.

Example

Finding Angle Measures Given an Arc Length

Finally, Paulina finished all her homework. She could now go to an amusement park with her friends. They get on a Ferris Wheel feeling super excited. The diagram indicates their positions.

Ferris wheel with the positions of friends' cabins
External credits: @wirestock

If the length of an arc between each cabin is 3 meters and the radius of the Ferris wheel is 12 meters, what is the measure of the angle formed by Paulina, the center of rotation, and Tiffaniqua? What is the measure of an angle formed by Paulina, Ali, and Tiffaniqua?

Hint
Identify the type of each angle. Use the formula for the arc length.

Solution
It is given that the length of an arc between each cabin is 3 meters. There is one cabin between Paulina and Tiffaniqua, so the arc between their cabins is 6 meters long.

Ferris wheel with the positions of friends and labeled arc length and radius

The length s of an arc of a circle with radius r can be calculated by using the following formula. s=2π r (θ/360^(∘)) Here, θ is the measure of the central angle, in degrees, that intercepts the arc. In this case, this is the angle formed by Paulina, the center of rotation, and Tiffaniqua. By substituting s= 6 and r= 12, its measure can be calculated.

s=2π r (θ/360^(∘))
SubstituteII

s= 6, r= 12

6=2π ( 12) (θ/360^(∘))
▼
Solve for θ
Multiply

Multiply

6=24π(θ/360^(∘))
MoveLeftFacToNum

a*b/c= a* b/c

6=24πθ/360^(∘)
ReduceFrac

a/b=.a /24./.b /24.

6=πθ/15^(∘)
MultEqn

LHS * 15^(∘)=RHS* 15^(∘)

90^(∘)=πθ
RearrangeEqn

Rearrange equation

πθ=90^(∘)
DivEqn

.LHS /π.=.RHS /π.

θ=90^(∘)/π
UseCalc

Use a calculator

θ = 28.647889...^(∘)
RoundDec

Round to 1 decimal place(s)

θ ≈ 28.6^(∘)

The measure of the angle formed between Paulina, the center of the Ferris wheel, and Tiffaniqua is about 28.6^(∘). This information can be used to find the measure of the angle formed by Paulina, Ali, and Tiffaniqua.

This angle, labeled as ∠ 1, is an inscribed angle that intercepts the arc between Paulina and Tiffaniqua. The same arc is intercepted by the central angle found earlier. Therefore, the measure of ∠ 1 is half the measure of that central angle. m∠ 1=28.6^(∘)/2 ⇕ m∠ 1=14.3^(∘) The measure of the angle formed by Paulina, Ali, and Tiffaniqua is about 14.3^(∘).
Closure

Finding the Orbit of a Satellite

The challenge presented at the beginning of this lesson can now be solved thanks to the concepts covered.

In 2007, a satellite named SELENE was launched to explore the Moon's surface for a few years. It orbited the Moon in a circular path staying at a constant altitude of 100 kilometers. Given that the radius of the Moon is 1737 kilometers, find the distance that the satellite traveled when it completed 60 % of its orbit.

The Moon and a satellite
External credits: @wirestock

Round the answer to the closest integer.

Hint
Start by calculating the measure of the corresponding arc. How can the radius of the orbit be calculated?

Solution
A full orbit around the moon is the equivalent to one full circle, which has a measure of 2π radians. By multiplying this value by 0.6, the measure of the arc that corresponds to 60 % of a circumference can be found. 0.6* 2π =1.2π This relationship can be illustrated in the diagram.

The Moon and a satellite
External credits: @wirestock

By the Segment Addition Postulate, the radius of the orbit is equal to the sum of the Moon's radius, which is 1737 kilometers, and 100 kilometers. r=1737+100 ⇔ r=1837km To find the length of the desired arc, the following formula can be used. s=θ r Here, θ is the measure, in radians, of the corresponding central angle. Recall that the measure of an arc is equal to the measure of its corresponding central angle. Therefore, θ equals 1.2π. By substituting θ= 1.2π and r= 1837 into the formula, the value of s can be calculated.

s=θ r
SubstituteII

θ= 1.2π, r= 1837

s=( 1.2π)( 1837)
UseCalc

Use a calculator

s=6925.326845...
RoundInt

Round to nearest integer

s≈ 6925

It has been determined that the distance traveled by the satellite when it completed 60 % of its orbit is 6925 kilometers.



Level 1
Level 2
Level 3
Radian Measure and Arc Length
Exercise 2.1
Use the unit circle to find the value of the following trigonometric expression where the argument is given in radians. Give the answer in exact form. sin π/4

sin 4π/3

To determine the value of sin π4, we will first convert the angle measure from radians to degrees. To do that, we will use the following formula. 180^(∘)=π By dividing this formula by 4, we can determine the corresponding angle of rotation in degrees.

A measure of π4 radians is equivalent to a rotation of 45^(∘). This is a special angle with a sine value of exactly 1sqrt(2)


As done in Part A, we will first convert the radians to degrees in order to determine which special angle corresponds to the given measurement.

As we can see, 4π3 radians corresponds to a rotation of 240^(∘). The reference angle of this rotation is 60^(∘).

The angle 60^(∘) is a special angle with a sine value of sqrt(3)2. However, notice that 240^(∘) is in the third quadrant where the y-values are negative. That means the value of sin 4π3 is the opposite number of sqrt(3)2.

As we can see, sin 4π3 =- sqrt(3)2.

E
Hint & Answer
S
Solution
In the diagram, we see an angle that measures 6 radians.

a unit circle with an angle that measures 6 radians
 Approximate how many degrees 6 radians is without the use of a calculator. Give the answer in degrees rounded to the nearest whole number.

Use the diagram to estimate the value of sin 6. Round the answer to one decimal place.

A full turn in the unit circle equals a circumference of 2π which approximates to 6.27 units. Therefore, 6 radians must be equal to an angle that is slightly less than a full turn.

To approximate 6 radians, let's view it together with an angle of 330^(∘).

The angle of 6 radians looks like it is halfway between 330^(∘) and 360^(∘). This means we can approximate how many degrees 6 radians corresponds to by adding these angles and dividing the sum by 2. 6 rad ≈ 330^(∘)+360^(∘)/2 = 345^(∘) Answers within an interval of 340^(∘) and 350^(∘) are acceptable

To estimate sin 6, recall that sine is given by the unit circle's vertical axis. Therefore, to find the sine value we must draw a horizontal line through the point on the unit circle that spans 6 radians of the circumference.

From the diagram, we see that 6 radians corresponds to a sine value of about - 0.3.

E
Hint & Answer
S
Solution

For which angle is the number of radians and degrees the same?

We want to investigate if there is any angle which has the same value in degrees and radians. If we assume such an angle has a measure in degrees of v, then the corresponding number of radians should be the following. v = v( π180) rad Let's solve this equation for v.

When v=0^(∘), the value is the same for radians and degrees.

E
Hint & Answer
S
Solution

A 75^(∘) arc in circle A and a 30^(∘) arc in circle B have the same length. What is the ratio of the radius r_A to the radius r_B?

Let's recall the formula for the length of an arc. s=2π r(θ/360^(∘)) We know that circle A has a sector with a central angle of 75^(∘) while the sector of circle B has a central angle of 30^(∘). Since the lengths of the arcs in the two circles are the same, we can write the following equation. 2π r_A(75^(∘)/360^(∘))=2π r_B(30^(∘)/360^(∘)) Let's simplify the equation. Our goal is to have the ratio of r_A to r_B on one side of the equation.

The ratio of the radius of circle A to the radius of circle B is 25.

E
Hint & Answer
S
Solution

Radian Measure and Arc Length

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