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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.

Round the answer to the closest integer.

b Find the measure of arc

A B .

A circle with an inscribed angle that measures 35 degrees and intercepts arc AB

c Determine the measures of

∠ S O R

and

∠ S T R .

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.

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^{\circ} .

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.

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

2 π

radii.

In calculations, even if the angle is given in radians, rad is seldom written. Instead, no unit marker indicates radians. Consider two expressions.

cos 6 4^{\circ} 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 for the measure of an angle. Knowing how to convert between them is useful, especially in trigonometry. Recall that a full circle measures

36 0^{\circ},

which corresponds to

2 π

radians.

36 0^{\circ} = 2 π rad \Leftrightarrow 18 0^{\circ} = π rad

Using this equality, it is possible to find equivalent expressions for

1^{\circ}

and

1 rad .

Degrees to Radians	Radians to Degrees
$1^{\circ} = \frac{π}{1 8 0} rad$	$1 rad = \frac{1 8 0 ^{\circ}}{π}$

Proof

The arc length for one revolution around a circle is also known as the circumference of the circle,

2 π r .

By dividing this with the length of the arc corresponding to one radian,

r,

the number of radians for one revolution is obtained.

\frac{2 π r}{r} = 2 π

This means that one revolution is

2 π

radians. In degrees, this is

36 0^{\circ},

leading to the relation

36 0^{\circ} = 2 π rad .

Dividing both sides by

2

gives the following equation.

18 0^{\circ} = π rad

Therefore,

18 0^{\circ}

correspond to

π

radians.

From the relation

18 0^{\circ} = π rad,

it is possible to find two rules, by dividing both sides by either

180

or

π .

To find an expression for

1^{\circ},

divide both sides by

180 .

18 0^{\circ} = π rad

DivEqn

$LHS / 180 = RHS / 180$

\frac{1 8 0}{1 8 0}^{\circ} = \frac{π}{1 8 0} rad

CalcQuot

Calculate quotient

1^{\circ} = \frac{π}{1 8 0} rad

As shown,

1^{\circ}

corresponds

\frac{π}{1 8 0},

which is approximately

0.017

radians. To get an expression for

1 rad,

divide both sides by

π .

18 0^{\circ} = π rad

DivEqn

$LHS / π = RHS / π$

\frac{1 8 0 ^{\circ}}{π} = \frac{π}{π} rad

CalcQuot

Calculate quotient

\frac{1 8 0 ^{\circ}}{π} = 1 rad

RearrangeEqn

Rearrange equation

1 rad = \frac{1 8 0 ^{\circ}}{π}

Therefore,

1

radian corresponds to

\frac{1 8 0}{π},

which is approximately

57 . 3^{\circ} .

A measure expressed in degrees can be converted into radians by multiplying by

\frac{π}{1 8 0} .

Similarly, multiplying a measure expressed in radians by

\frac{1 8 0 ^{\circ}}{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.

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.

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.

\frac{r _{1}}{r _{2}} = \frac{s _{1}}{s _{2}}

This proportion can be rewritten as an equivalent equation.

\frac{r _{1}}{r _{2}} = \frac{s _{1}}{s _{2}}

▼

Rearrange equation

MultEqn

$LHS \cdot r_{2} s_{2} = RHS \cdot r_{2} s_{2}$

\frac{r _{1}}{r _{2}} \cdot r_{2} s_{2} = \frac{s _{1}}{s _{2}} \cdot r_{2} s_{2}

CancelCommonFac

Cancel out common factors

\frac{r _{1}}{r _{2}} \cdot r_{2} s_{2} = \frac{s _{1}}{s _{2}} \cdot r_{2} s_{2}

SimpQuotProd

Simplify quotient and product

r_{1} s_{2} = s_{1} r_{2}

DivEqn

$LHS / r_{1} = RHS / r_{1}$

s_{2} = \frac{s _{1} r _{2}}{r _{1}}

MovePartNumRight

$\frac{a \cdot b}{c} = \frac{a}{c} \cdot b$

s_{2} = \frac{s _{1}}{r _{1}} r_{2}

DivEqn

$LHS / r_{2} = RHS / r_{2}$

\frac{s _{2}}{r _{2}} = \frac{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.

$θ = \frac{s}{r}$

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

\frac{π}{1 8 0 ^{\circ}},

the measure can be converted into radians and substituted into the formula.

s = (\frac{π}{1 8 0 ^{\circ}} θ) r ⇕ s = π r (\frac{θ}{1 8 0 ^{\circ}})

This formula is often written in the following equivalent manner.

$s = 2 π r (\frac{θ}{3 6 0 ^{\circ}})$

This equivalent form is convenient to work with because $2 π r$ is the circumference of a circle. Since a full circle measures $36 0^{\circ},$ dividing $2 π r$ 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 $2 P . M .$ She calculated the angle between the longer and shorter hands of the clock to be $6 0^{\circ} .$

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 $s = 2 π r (\frac{θ}{3 6 0 ^{\circ}}) .$

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 = 2 π r (\frac{θ}{3 6 0 ^{\circ}})

It is known that the radius of the clock is about

15

centimeters and the central angle that intercepts the arc measures

6 0^{\circ} .

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

s = 2 π r (\frac{θ}{3 6 0 ^{\circ}})

SubstituteII

$r = 15$ , $θ = 6 0^{\circ}$

s = 2 π (15) (\frac{6 0 ^{\circ}}{3 6 0 ^{\circ}})

▼

Evaluate right-hand side

ReduceFrac

$\frac{a}{b} = \frac{a / 6 0 ^{\circ}}{b / 6 0 ^{\circ}}$

s = 2 π (15) (\frac{1}{6})

MoveLeftFacToNumOne

$a \cdot \frac{1}{b} = \frac{a}{b}$

s = \frac{2 π ( 1 5 )}{6}

Multiply

s = \frac{3 0 π}{3}

ReduceFrac

$\frac{a}{b} = \frac{a / 3}{b / 3}$

s = 10 π

UseCalc

Use a calculator

s = 31.415926 \dots

RoundDec

Round to $1$ decimal place(s)

s \approx 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 $4 0^{\circ} .$

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

M N .

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 $\frac{π}{1 8 0 ^{\circ}} .$ 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

∠ M K N,

which is an inscribed angle, measures

4 0^{\circ} .

By multiplying this value by the conversion factor

\frac{π}{1 8 0 ^{\circ}},

the measure of this angle can be converted from degrees to radians.

m ∠ M K N = 4 0^{\circ} \cdot \frac{π}{1 8 0 ^{\circ}}

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

m ∠ M K N = \frac{4 0 ^{\circ} π}{1 8 0 ^{\circ}}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 0 ^{\circ}}{b / 2 0 ^{\circ}}$

m ∠ M K N = \frac{2 π}{9}

Next, to find the length of

M N,

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,

∠ M K N

corresponds to a central angle

∠ M O N .

Using this information, the measure of

∠ M O N

can be found.

m ∠ M K N = \frac{1}{2} m ∠ M O N

Substitute

$m ∠ M K N = \frac{2 π}{9}$

\frac{2 π}{9} = \frac{1}{2} m ∠ M O N

▼

Solve for

m ∠ M O N

MultEqn

$LHS \cdot 2 = RHS \cdot 2$

\frac{2 π}{9} (2) = m ∠ M O N

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

\frac{2 π ( 2 )}{9} = m ∠ M O N

Multiply

\frac{4 π}{9} = m ∠ M O N

RearrangeEqn

Rearrange equation

m ∠ M O N = \frac{4 π}{9}

Finally, the length of

M N

can be calculated using the corresponding formula.

s = θ r ⇓ M N = m ∠ M O N r

Here,

\frac{4 π}{9}

and

5

can be substituted for

m ∠ M O N

and

r,

respectively.

M N = m ∠ M O N r

SubstituteII

$m ∠ M O N = \frac{4 π}{9}$ , $r = 5$

M N = (\frac{4 π}{9}) (5)

▼

Evaluate right-hand side

MoveRightFacToNum

$\frac{a}{c} \cdot b = \frac{a \cdot b}{c}$

M N = \frac{4 π ( 5 )}{9}

Multiply

M N = \frac{2 0 π}{9}

UseCalc

Use a calculator

M N = 6.981317 \dots

RoundInt

Round to nearest integer

M N \approx 7

The length of

M N

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.

The length

s

of an arc of a circle with radius

r

can be calculated by using the following formula.

s = 2 π r (\frac{θ}{3 6 0 ^{\circ}})

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 = 2 π r (\frac{θ}{3 6 0 ^{\circ}})

SubstituteII

$s = 6$ , $r = 12$

6 = 2 π (12) (\frac{θ}{3 6 0 ^{\circ}})

▼

Solve for

θ

Multiply

6 = 24 π (\frac{θ}{3 6 0 ^{\circ}})

MoveLeftFacToNum

$a \cdot \frac{b}{c} = \frac{a \cdot b}{c}$

6 = \frac{2 4 π θ}{3 6 0 ^{\circ}}

ReduceFrac

$\frac{a}{b} = \frac{a / 2 4}{b / 2 4}$

6 = \frac{π θ}{1 5 ^{\circ}}

MultEqn

$LHS \cdot 1 5^{\circ} = RHS \cdot 1 5^{\circ}$

9 0^{\circ} = π θ

RearrangeEqn

Rearrange equation

π θ = 9 0^{\circ}

DivEqn

$LHS / π = RHS / π$

θ = \frac{9 0 ^{\circ}}{π}

UseCalc

Use a calculator

θ = 28.647889 \dots^{\circ}

RoundDec

Round to $1$ decimal place(s)

θ \approx 28 . 6^{\circ}

The measure of the angle formed between Paulina, the center of the Ferris wheel, and Tiffaniqua is about

28 . 6^{\circ} .

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 = \frac{2 8 . 6 ^{\circ}}{2} ⇕ m ∠ 1 = 14 . 3^{\circ}

The measure of the angle formed by Paulina, Ali, and Tiffaniqua is about

14 . 3^{\circ} .

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.

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

2 π

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 \cdot 2 π = 1.2 π

This relationship can be illustrated in the diagram.

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 \Leftrightarrow r = 1837 km

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

1.2 π .

By substituting

θ = 1.2 π

and

r = 1837

into the formula, the value of

s

can be calculated.

s = θ r

SubstituteII

$θ = 1.2 π$ , $r = 1837$

s = (1.2 π) (1837)

UseCalc

Use a calculator

s = 6925.326845 \dots

RoundInt

Round to nearest integer

s \approx 6925

It has been determined that the distance traveled by the satellite when it completed

60 %

of its orbit is

6925

kilometers.

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Catch-Up and Review

Proof

Hint

Solution

Hint

Solution

Hint

Solution

Hint

Solution