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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.

A circle is the set of all the points in a plane that are equidistant from a given point. There are a few particularly notable features of a circle.

- Center - The given point from which all points of the circle are equidistant. Circles are often named by their center point.
- Radius - A segment that connects the center and any point on the circle. Its length is usually represented algebraically by $r.$
- Diameter - A segment whose endpoints are on the circle and passes through the center. Its length is usually represented algebraically by $d.$
- Circumference - The perimeter of a circle, usually represented algebraically by $C.$

circle $O.$

In any given circle, the lengths of any radius and any diameter are constant. They are called

The circumference of a circle is calculated by multiplying its diameter by $π.$

$C=πd$

Since the diameter is twice the radius, the circumference of a circle can also be calculated by multiplying $2r$ by $π.$

$C=2πr$

Consider two circles and their respective diameters and circumferences.

By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional. $C_{A}C_{B} =d_{A}d_{B} $ This proportion can be rearranged.$C_{A}C_{B} =d_{A}d_{B} $

Rearrange equation

MultEqn

$LHS⋅C_{A}=RHS⋅C_{A}$

$C_{B}=d_{A}d_{B} (C_{A})$

DivEqn

$LHS/d_{B}=RHS/d_{B}$

$d_{B}C_{B} =d_{A}1 (C_{A})$

MoveRightFacToNumOne

$b1 ⋅a=ba $

$d_{B}C_{B} =d_{A}C_{A} $

$dC =π⇒C=πd$

An angle whose vertex lies at the center of a circle and whose legs are radii is called a central angle.

In this diagram, $∠C$ is a central angle of the circle. Note that the measure of the intercepted arc of a central angle is equal to the measure of the central angle.

Recall that both a circle and a complete angle measure $360$ degrees. Since the vertex of a central angle lies at the center of a circle, both the central angle and its intercepted arc represent the same portion of the circle. Use the slider in the applet to see this relationship.

Therefore, the central angle and its intercepted arc have the same measure.

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 $AB.$

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 $ADB.$ 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 its corresponding central angle. In the diagram, the measure of $AB$ is equal to the measure of $∠ACB.$

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

An arc length is a portion of the circumference of a circle.

In the diagram, the measure of $AB$ equals $θ,$ while the length of $AB$ corresponds to the portion of the circumference the red arc spans. To find it, the relationship between arc length and arc measure can be used.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 $

SubstituteValues

Substitute values

$4π1.22 =360_{∘}arc measure $

UseCalc

Use a calculator

$0.09708…=360_{∘}arc measure $

MultEqn

$LHS⋅360_{∘}=RHS⋅360_{∘}$

$34.95_{∘}≈arc measure$

RearrangeEqn

Rearrange 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 $

SubstituteValues

Substitute values

$8πarc length =360_{∘}50_{∘} $

CalcQuot

Calculate quotient

$8πarc length =0.13888…$

MultEqn

$LHS⋅8π=RHS⋅8π$

$arc length≈3.49$

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 the *constant of proportionality,* and defines the radian measure of the central angle associated with the arc. Because the circumference of a circle with radius $1$ is $2π,$ the radian measure of a circle is $2π$ radians — which corresponds to $360_{∘}.$

Because 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 $

$180_{∘}πradians⋅45_{∘} =x$

SimpQuot

Simplify quotient

$4πradians =x$

UseCalc

Use a calculator

$0.78539…radians=x$

RearrangeEqn

Rearrange equation

$x=0.78539…radians$

The area of a circle is the product of $π$ and the square of its radius.

A circle with radius $r$ will be divided into a number of equally sized sectors. Then, the top and bottom halves of the circle will be distinguished by filling them with different colors. Because the circumference of a circle is $2πr,$ the arc length of each semicircle is half this value, $πr.$

Now, the above sectors will be *unfolded*. By placing the sectors of the upper hemisphere as teeth pointing downwards and the sectors of the bottom hemisphere as teeth pointing upwards, a parallelogram-like figure can be formed. Therefore, the area of the figure below should be the same as the circle's area.

Here, the shorter sides become

$A=πr⋅r⇔A=πr_{2}$

It was shown that the area of a circle is the product of $π$ and the square of its radius.

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

In the diagram, sector $ACB$ is created by $AC,BC,$ and $AB.$

The area of a sector of a circle is calculated by multiplying the circle's area by the ratio of the measure of the central angle to $360_{∘}.$

$Area of Sector=360_{∘}θ ⋅πr_{2}$

From the fact that $2πrad$ equals $360_{∘},$ an equivalent formula can be written if the central angle is given in radians.

$Area of SectorArea of Sector =2πθ ⋅πr_{2}⇓=2θ ⋅r_{2} $

Since the measure of an arc is equal to the measure of its central angle, the arc $AB$ measures $θ.$ Therefore, by substituting $mAB$ for $θ,$ another version of the formula is obtained which can also be written in degrees or radians.

$Area of Sector=360_{∘}mAB ⋅πr_{2}$

or

$Area of Sector=2mAB ⋅r_{2}$

Consider sector $ACB$ bounded by $AC,BC,$ and $AB.$

Since a circle measures $360_{∘},$ this sector represents $360_{∘}θ $ of $⊙C.$ Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to $360_{∘}θ .$ $Area of CircleArea of Sector =360_{∘}θ $

Recall that the area of a circle is $πr_{2}.$ By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained. $πr_{2}Area of Sector =360_{∘}θ $ Therefore, the area of a sector of a circle can be found by 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 $125.66cm_{2}.$

$area of sector=360_{∘}θ ⋅πr_{2}$

SubstituteII

$θ=100_{∘}$, $r=12$

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

UseCalc

Use a calculator

$area of sector≈125.66$

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