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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.

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b Find the measure of arc $AB.$

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.02533em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.02533em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\">A<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05017em;\">B<\/span><\/span><\/span><span class=\"svg-align\" style=\"top:-3.6833299999999998em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"stretchy\" style=\"height:0.342em;min-width:0.888em;\"><span class=\"halfarrow-left\" style=\"height:0.342em;\"><svg width='400em' height='0.342em' viewBox='0 0 400000 342' preserveAspectRatio='xMinYMin slice'><path d='M400000 80\nH435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0\n 435 0h399565z'\/><\/svg><\/span><span class=\"halfarrow-right\" style=\"height:0.342em;\"><svg width='400em' height='0.342em' viewBox='0 0 400000 342' preserveAspectRatio='xMaxYMin slice'><path d='M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0\n 3-1 3-3v-38c-76-158-257-219-435-219H0z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["70"]}}

c Determine the measures of $∠SOR$ and $∠STR.$

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.02778em;\">O<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["104"]}}

{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.69224em;vertical-align:0em;\"><\/span><span class=\"mord mathdefault\">m<\/span><span class=\"mord\">\u2220<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.05764em;\">S<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.13889em;\">T<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.00773em;\">R<\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.674115em;vertical-align:0em;\"><\/span><span class=\"mord\"><span><\/span><span class=\"msupsub\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:0.674115em;\"><span style=\"top:-3.063em;margin-right:0.05em;\"><span class=\"pstrut\" style=\"height:2.7em;\"><\/span><span class=\"sizing reset-size6 size3 mtight\"><span class=\"mbin mtight\">\u2218<\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span><\/span>","answer":{"text":["52"]}}

Challenge

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

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 $57.3_{∘}.$

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 $2π$ radii.

In calculations, even if the angle is given in radians,

radis seldom written. Instead, no unit marker indicates radians. Consider two expressions.

$cos64_{∘}andcos5 $

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

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 $360_{∘},$ which corresponds to $2π$ radians.

### Proof

$360_{∘}=2πrad⇔180_{∘}=πrad $

Using this equality, it is possible to find equivalent expressions for $1_{∘}$ and $1rad.$ Degrees to Radians | Radians to Degrees |
---|---|

$1_{∘}=180π rad$ | $1rad=π180_{∘} $ |

The arc length for one revolution around a circle is also known as the circumference of the circle, $2πr.$ By dividing this with the length of the arc corresponding to one radian, $r,$ the number of radians for one revolution is obtained.
As shown, $1_{∘}$ corresponds $180π ,$ which is approximately $0.017$ radians. To get an expression for $1rad,$ divide both sides by $π.$
Therefore, $1$ radian corresponds to $π180 ,$ which is approximately $57.3_{∘}.$

$r2πr =2π $

This means that one revolution is $2π$ radians. In degrees, this is $360_{∘},$ leading to the relation $360_{∘}=2πrad.$ Dividing both sides by $2$ gives the following equation.
$180_{∘}=πrad $

Therefore, $180_{∘}$ correspond to $π$ radians.
From the relation $180_{∘}=πrad,$ 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 $180.$
$180_{∘}=πrad$

DivEqn

$LHS/180=RHS/180$

$180180 _{∘}=180π rad$

CalcQuot

Calculate quotient

$1_{∘}=180π rad$

$180_{∘}=πrad$

DivEqn

$LHS/π=RHS/π$

$π180_{∘} =ππ rad$

CalcQuot

Calculate quotient

$π180_{∘} =1rad$

RearrangeEqn

Rearrange equation

$1rad=π180_{∘} $

Pop Quiz

Pop Quiz

Explore

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. *radius* changes.

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

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

Discussion

To understand the observed relation, consider two concentric circles with different radii $r_{1}$ and $r_{2}.$

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_{2}r_{1} =s_{2}s_{1} $

This proportion can be rewritten as an equivalent equation.
$r_{2}r_{1} =s_{2}s_{1} $

▼

Rearrange equation

MultEqn

$LHS⋅r_{2}s_{2}=RHS⋅r_{2}s_{2}$

$r_{2}r_{1} ⋅r_{2}s_{2}=s_{2}s_{1} ⋅r_{2}s_{2}$

CancelCommonFac

Cancel out common factors

$r_{2} r_{1} ⋅r_{2} s_{2}=s_{2} s_{1} ⋅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}=r_{1}s_{1}r_{2} $

MovePartNumRight

$ca⋅b =ca ⋅b$

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

DivEqn

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

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

Discussion

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.

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

$θ=rs $

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

$s=θr$

$s=(180_{∘}π θ)r⇕s=πr(180_{∘}θ ) $

This formula is often written in the following equivalent manner. $s=2πr(360_{∘}θ )$

This equivalent form is convenient to work with because $2πr$ is the circumference of a circle. Since a full circle measures $360_{∘},$ dividing $2πr$ 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.

Example

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":["x"],"constants":["PI"]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:0.9580078125em;vertical-align:-0.2080078125em;\"><\/span><span class=\"mord text\"><span class=\"mord Roboto-Regular\">Arc<\/span><span class=\"mord\">\u00a0<\/span><span class=\"mord Roboto-Regular\">Length<\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"cm","answer":{"text":["31.4"]}}

Since the central angle is given in degrees, the arc length can be calculated using the formula $s=2πr(360_{∘}θ ).$

To find the length of an arc using the measure of the corresponding central angle given in degrees, the following formula can be used.
The arc between the hands of the clock is about $31.4$ centimeters long.

$s=2πr(360_{∘}θ ) $

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.
$s=2πr(360_{∘}θ )$

SubstituteII

$r=15$, $θ=60_{∘}$

$s=2π(15)(360_{∘}60_{∘} )$

▼

Evaluate right-hand side

ReduceFrac

$ba =b/60_{∘}a/60_{∘} $

$s=2π(15)(61 )$

MoveLeftFacToNumOne

$a⋅b1 =ba $

$s=62π(15) $

Multiply

Multiply

$s=330π $

ReduceFrac

$ba =b/3a/3 $

$s=10π$

UseCalc

Use a calculator

$s=31.415926…$

RoundDec

Round to $1$ decimal place(s)

$s≈31.4$

Example

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 $40_{∘}.$

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.{"type":"text","form":{"type":"math","options":{"comparison":"1","nofractofloat":false,"keypad":{"simple":true,"useShortLog":false,"variables":[],"constants":[]}},"text":"<span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><\/span><\/span>"},"formTextBefore":"Length of <span class=\"katex\"><span class=\"katex-html\" aria-hidden=\"true\"><span class=\"base\"><span class=\"strut\" style=\"height:1.02533em;vertical-align:0em;\"><\/span><span class=\"mord accent\"><span class=\"vlist-t\"><span class=\"vlist-r\"><span class=\"vlist\" style=\"height:1.02533em;\"><span style=\"top:-3em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"mord\"><span class=\"mord mathdefault\" style=\"margin-right:0.10903em;\">M<\/span><span class=\"mord mathdefault\" style=\"margin-right:0.10903em;\">N<\/span><\/span><\/span><span class=\"svg-align\" style=\"top:-3.6833299999999998em;\"><span class=\"pstrut\" style=\"height:3em;\"><\/span><span class=\"stretchy\" style=\"height:0.342em;min-width:0.888em;\"><span class=\"halfarrow-left\" style=\"height:0.342em;\"><svg width='400em' height='0.342em' viewBox='0 0 400000 342' preserveAspectRatio='xMinYMin slice'><path d='M400000 80\nH435C64 80 168.3 229.4 21 260c-5.9 1.2-18 0-18 0-2 0-3-1-3-3v-38C76 61 257 0\n 435 0h399565z'\/><\/svg><\/span><span class=\"halfarrow-right\" style=\"height:0.342em;\"><svg width='400em' height='0.342em' viewBox='0 0 400000 342' preserveAspectRatio='xMaxYMin slice'><path d='M0 80h399565c371 0 266.7 149.4 414 180 5.9 1.2 18 0 18 0 2 0\n 3-1 3-3v-38c-76-158-257-219-435-219H0z'\/><\/svg><\/span><\/span><\/span><\/span><\/span><\/span><\/span><span class=\"mspace\" style=\"margin-right:0.2777777777777778em;\"><\/span><span class=\"mrel\">=<\/span><\/span><\/span><\/span>","formTextAfter":"inches","answer":{"text":["7"]}}

To convert the measure of an angle from degrees to radians, use the conversion factor $180_{∘}π .$ The length of the arc can be found by using the formula $s=θr,$ where $θ$ is the measure of the corresponding central angle in radians.

It can be seen in the diagram that $∠MKN,$ which is an inscribed angle, measures $40_{∘}.$
By multiplying this value by the conversion factor $180_{∘}π ,$ the measure of this angle can be converted from degrees to radians.
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 $∠MON.$
Using this information, the measure of $∠MON$ can be found.
Finally, the length of $MN$ can be calculated using the corresponding formula.
The length of $MN$ is approximately $7$ inches.

$m∠MKN=40_{∘}⋅180_{∘}π $

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$m∠MKN=180_{∘}40_{∘}π $

ReduceFrac

$ba =b/20_{∘}a/20_{∘} $

$m∠MKN=92π $

$m∠MKN=21 m∠MON$

Substitute

$m∠MKN=92π $

$92π =21 m∠MON$

▼

Solve for $m∠MON$

MultEqn

$LHS⋅2=RHS⋅2$

$92π (2)=m∠MON$

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$92π(2) =m∠MON$

Multiply

Multiply

$94π =m∠MON$

RearrangeEqn

Rearrange equation

$m∠MON=94π $

$s=θr⇓MN=m∠MONr $

Here, $94π $ and $5$ can be substituted for $m∠MON$ and $r,$ respectively.
$MN=m∠MONr$

SubstituteII

$m∠MON=94π $, $r=5$

$MN=(94π )(5)$

▼

Evaluate right-hand side

MoveRightFacToNum

$ca ⋅b=ca⋅b $

$MN=94π(5) $

Multiply

Multiply

$MN=920π $

UseCalc

Use a calculator

$MN=6.981317…$

RoundInt

Round to nearest integer

$MN≈7$

Example

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.

External credits: @wirestock

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

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=2πr(360_{∘}θ ) $

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.
$s=2πr(360_{∘}θ )$

SubstituteII

$s=6$, $r=12$

$6=2π(12)(360_{∘}θ )$

▼

Solve for $θ$

Multiply

Multiply

$6=24π(360_{∘}θ )$

MoveLeftFacToNum

$a⋅cb =ca⋅b $

$6=360_{∘}24πθ $

ReduceFrac

$ba =b/24a/24 $

$6=15_{∘}πθ $

MultEqn

$LHS⋅15_{∘}=RHS⋅15_{∘}$

$90_{∘}=πθ$

RearrangeEqn

Rearrange equation

$πθ=90_{∘}$

DivEqn

$LHS/π=RHS/π$

$θ=π90_{∘} $

UseCalc

Use a calculator

$θ=28.647889…_{∘}$

RoundDec

Round to $1$ decimal place(s)

$θ≈28.6_{∘}$

$m∠1=228.6_{∘} ⇕m∠1=14.3_{∘} $

The measure of the angle formed by Paulina, Ali, and Tiffaniqua is about $14.3_{∘}.$
Closure

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

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

$0.6⋅2π=1.2π $

This relationship can be illustrated in the diagram.
External credits: @wirestock

$r=1737+100⇔r=1837km $

To find the length of the desired arc, the following formula can be used.
$s=θr $

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.
$s=θr$

SubstituteII

$θ=1.2π$, $r=1837$

$s=(1.2π)(1837)$

UseCalc

Use a calculator

$s=6925.326845…$

RoundInt

Round to nearest integer

$s≈6925$

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