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

$180_{∘}=πrad $

Using this equality, the conversion factors from degrees to radians and from radians to degrees can be derived. Degrees to Radians | Radians to Degrees |
---|---|

$1_{∘}=180π rad$ | $1rad=π180_{∘} $ |

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} $

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.

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)

$s≈31.4$

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

$MN≈7$