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Circumference and area are two measurements that can be used to analyze circles. The first measures the boundary or perimeter of a circle, while the second measures the space inside a circle. Sometimes, only a portion of the circumference and/or area — *arcs* and *sectors,* respectively — are explored.

An angle whose vertex lies at the center of a circle is called a central angle.

In the diagram above, $∠C$ — also named $∠ACB$ — is a central angle of the circle.

Suppose $∠ACB$ is drawn in circle $C$ such that it is a central angle. If $m∠ACB$ is less than $180_{∘},$ the endpoints of the angle, $A$ and $B,$ create a minor arc on the circle. This arc can be named $\Arc{AB}.$

If $m∠ACB$ is *greater* than $180_{∘},$ the endpoints of the angle create a major arc on the circle. In other words, the points on the circle not included in the minor arc create a major arc.

A major arc can be named using its endpoints and another point on the arc. Therefore, the major arc can be named $\Arc{ADB}.$ In the diagram above,

- $C$ is the center of the circle,
- $∠ACB$ is a central angle of the circle,
- arc $AB$ is a minor arc, and
- arc $ADB$ is a major arc.

The measure of an arc is equal to the measure of the central angle that creates it. Suppose arc $AB$ is created when the central angle $∠ACB$ is drawn inside circle $C.$ In the diagram above, the measure of arc $AB=m∠C.$

Suppose the measure of arc $AB$ is given as $θ.$ The measure of the corresponding major arc, arc $ADB,$ can be found by subtracting $θ$ from $360_{∘},$ the measure of the entire circle.

Arc length measures the portion of the circle's circumference an arc spans. In the diagram below, the measure of arc $AB$ equals $θ,$ while the **length** of arc $AB$ corresponds to the portion of the circumference the red arc spans.

The ratio of an arc's length to the circumference of the circle is equal to the ratio of the arc's measure to $360_{∘}.$

$circumferencearc length =360_{∘}arc measure $

Substituting $θ$ for the arc measure and $2πr$ for the circumference gives the following.

$2πrarc length =360_{∘}θ $Find the measure of arcs $AB$ and $BCA.$

Show Solution

To begin, let's make sense of the given information. As can be seen, $1.22$ is the length of arc $AB,$ and $2$ is the radius of the circle. Because arc length is given and arc measure needs to be determined, we can use the relationship between the two.
$circumferencearc length =360_{∘}arc measure $
To find the measure of arc $AB,$ we must first calculate the circumference of the circle.
$CCC =2πr=2π⋅2=4π $
Now, we can substitute the known values for circumference and arc length into the equation to solve for the measure of arc $AB.$
Thus, the measure of arc $AB$ is approximately $34.95_{∘}.$ Notice that arc $AB$ is a minor arc and arc $BCA$ is its corresponding major arc. As a result, we can subtract $34.95_{∘}$ from $360_{∘}$ to find the measure of arc $BCA.$
$360_{∘}−34.95_{∘}=325.05_{∘}$
Thus, the measure of arc $BCA$ is approximately $325.05_{∘}.$

$circumferencearc length =360_{∘}arc measure $

SubstituteValuesSubstitute values

$4π1.22 =360_{∘}arc measure $

UseCalcUse a calculator

$0.09708…=360_{∘}arc measure $

MultEqn$LHS⋅360_{∘}=RHS⋅360_{∘}$

$34.95_{∘}≈arc measure$

RearrangeEqnRearrange equation

$arc measure≈34.95_{∘}$

Find the length of arc $AB.$

Show Solution

To begin, we can make sense of the given information. The circle $C$ has a radius of $4$ inches and the arc $AB$ mesaures $50_{∘}.$ The relationship between arc measure and arc length can be used. $circumferencearc length =360_{∘}arc measure $ First, the circumference of circle $C$ can be calculated as follows. $CCC =2πr=2π⋅4=8π $

The length of arc $AB$ can be calculated by substituting the values for circumference and arc measure, and solving.$circumferencearc length =360_{∘}arc measure $

SubstituteValuesSubstitute values

$8πarc length =360_{∘}50_{∘} $

CalcQuotCalculate quotient

$8πarc length =0.13888…$

MultEqn$LHS⋅8π=RHS⋅8π$

$arc length≈3.49$

In the diagram below, circle $AB$ has a radius of $1$ unit, and circle $CD$ has a radius of $r$ units.

The measures of arcs $AB$ and $CD$ are equal because they are created by the same central angle. Because circle $AB$ has a radius of $1,$ the length of arc $AB$ can be found using the definition of arc length. $arc lengthAB=360_{∘}arc measureAB ⋅2π$ Because all circles are similar, circle $AB∼$ circle $CD.$ Thus, the ratios of corresponding parts between the circles are proportional. An equation relating arc length to radius can be written. $arc lengthABarc lengthCD =1r $ Simplifying the right-hand side and substituting the found value for the length of arc $AB$ gives the following. $arc lengthCD =r⋅arc length AB=r⋅(360_{∘}arc measureAB ⋅2π) $ The equation above shows that the length of arc $CD$ is proportional to the radius of the circle. In fact, the expression $360_{∘}arc measureAB ⋅2π$

is called theBecause angles can be measured in both radians and degrees, it can be useful to translate between the two. The concept of radian measure asserts that one revolution around a circle corresponds to exactly $2π$ radians. It is also known that this same distance can be expressed as $360_{∘}.$ Thus, $2πradians=360_{∘}.$ Simplifying gives the following conversion factor.

$πradians=180_{∘}$

Convert $45_{∘}$ to radians and $2π radians$ to degrees.

Show Solution

To convert both measures, we can use the following relation.
$πradians=180_{∘}$
The following proportion can be written to convert $45_{∘}$ to radians.
$180_{∘}πradians =45_{∘}x $
Solving the equation for $x$ will give the corresponding value.
Thus, $45_{∘}$ corresponds to approximately $0.79$ radians. In a similar way, we can convert $2π $ radians to degrees.
$180_{∘}πradians =yπ/2 ⇔y=90_{∘}$
Thus, $2π $ radians corresponds to $90_{∘}.$

$180_{∘}πradians =45_{∘}x $

MultEqn$LHS⋅45_{∘}=RHS⋅45_{∘}$

$180_{∘}πradians ⋅45_{∘}=x$

MoveRightFacToNum$ca ⋅b=ca⋅b $

$180_{∘}πradians⋅45_{∘} =x$

SimpQuotSimplify quotient

$4πradians =x$

UseCalcUse a calculator

$0.78539…radians=x$

RearrangeEqnRearrange equation

$x=0.78539…radians$

A sector of a circle is a portion of the circle enclosed by two radii and an arc.

In the diagram, sector $ACB$ is created by $AC,BC$ and arc $AB.$The area of a sector of a circle can be determined using the proportional relationship between the circle's and the sector's areas and measures. $area of circlearea of sector =measure of circlemeasure of arcAB $ Substituting the known values gives the following. $πr_{2}area of sector =360_{∘}θ $ Therefore, the area of a sector of a circle can be found using the following formula.

$area of sector=360_{∘}θ ⋅πr_{2}$

Find the area of the sector.

Show Solution

To begin, notice that the arc corresponding with the given sector measures $100_{∘}.$ Notice also that the radius of the circle is $12$ cm. We can substitute these values into the formula for the area of a sector and solve.
Thus, the area of the sector is approximately $125.66cm_{2}.$

$area of sector=360_{∘}θ ⋅πr_{2}$

$area of sector=360_{∘}100_{∘} ⋅π⋅12_{2}$

UseCalcUse a calculator

$area of sector≈125.66$

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