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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 1.1

Convert the angle measurement from degrees to radians. Give the answer in exact form.

30^(∘)
15^(∘)
-105^(∘)
450^(∘)

To convert from degrees to radians, we use the following relationship. 180^(∘)=π rad By dividing both sides of this equation by 180^(∘), we get a relationship showing how many radians that corresponds to 1^(∘). 1^(∘)=π/180 rad Next, we can convert the given number of degrees to radians if we multiply both sides by 30.


We will use the same relationship as in Part A to convert 15^(∘) to radians.


Again, let's repeat the procedure from Part A and B.


Like in previous parts, we will repeat the established procedure to convert 450^(∘) to radians.


E
Hint & Answer
S
Solution

Convert the angle measurement from radians to degrees.

π/4 rad
π/3 rad
2π/3 rad
πrad

To convert from radians to degrees, we can use the following formula. π rad=180^(∘) By dividing both sides by π , we get a formula showing the number of degrees that correspond to 1 radian. 1 rad=180^(∘)/π Next, we can convert the angle measure from radians to degrees by multiplying both sides of this relationship by π4.


Let's use the same relationship as in Part A to convert π3 to degrees.


We will repeat the procedure from Part A and B.


Again, we will repeat the established procedure to convert π radians to degrees.


E
Hint & Answer
S
Solution

Zain and Tearrik both use their calculators to determine the sine value of 30^(∘). Wait a minute. They get different results.

Something must be going on here. Who is correct?

Tearrik has the correct answer. The error Zain made is that he set his calculator to use radians instead of degrees. If we determine sin 30 with the calculator set to radians, we would obtain - 0.9880316241. That is the answer which Zain calculated.

To change from radians to degrees, push MODE and then select Degree on the third row.

E
Hint & Answer
S
Solution

What is the ratio of the length of the arc to the radius of the circle. Answer in exact form.

a circle with a radius of 2 and a sector that spans 60 degrees

What is the ratio of the length of the arc to the radius of the circle. Answer in exact form.

a circle with a radius of 4 and a sector that also spans 60 degrees
 The ratio of the arc length to the radius of a circle is called a radian. Will all sectors with a central angle of 60^(∘) have the same radian?

To find the arc length of AB we have to multiply the circumference of the circle with the ratio of the central angle to 360^(∘).

The arc length is 2π3. Finally, we will divide the arc length with the radius of D which is 2. .2π /3./2=2π/2(3)=π/3

As in Part A, we will find the arc length by multiplying the circumference of the circle with the ratio of the central angle to 360^(∘)

The arc length is 4π3. As in Part A, we will divide the arc length with the radius of D. .4π /3./4=4π/4(3)=π/3

To show that the radian measure of a sector that spans 60^(∘) always is π3 we will calculate the sector length for a circle with a radius of x units.

Let's calculate the arc length.

The arc length is xπ3. As in Part A, we will divide the arc length with the radius of ⊙ C. .π x /3./x=π x/3x=π/3 As we can see, the ratio is π3 regardless of the radius of the circle.

E
Hint & Answer
S
Solution
The radian measure of a central angle of 90^(∘) is π2. Find the radian measure of a central angle of 180^(∘) without the use of a calculator.

The angle measure in radian is the arc length divided by the radius of a circle. Radian= Arc length/Radius If we increase the central angle from 90^(∘) to 180^(∘), we have doubled the arc length. This must also mean that the radian measure of the central angle has doubled. 2(π/2)=π Therefore, the radian measure will equal π.

E
Hint & Answer
S
Solution

Calculate the value of the trigonometric expression where the argument is given in radians. Round the value to three decimal places.

sin 4
sin 10

The value of sin 4 can not be expressed exactly. This means we must use a calculator to determine this. First, make sure that the calculator is set to radians. Push MODE and select Radian on the third row.

The calculator is now set to interpret the argument of a trigonometric expressions in radians. If we write sin 4 on the calculator, the result will be the sine value of 4 radians.

When we round to three decimal places, we see that sin 4 has a value of - 0.757.

As in Part A, we need a calculator to determine sin 10. Since the calculator is already set to radians, we can immediately determine the value of sin 10.

When we round to three decimal places, sin 10 has a value of - 0.544.

E
Hint & Answer
S
Solution

Radian Measure and Arc Length

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