Using Circumference and Area

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.

Rule

Circumference of a Circle

The circumference is the distance around a circle.

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

$C = π d$

Since the diameter is twice the length of the radius, the circumference can also be expressed as:

$C = 2 π r .$

Concept

Central Angle

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

In the diagram above,

∠ C

— also named

∠ A C B

— is a central angle of the circle.

Concept

Minor and Major Arc

Suppose $∠ A C B$ is drawn in circle $C$ such that it is a central angle. If $m ∠ A C B$ is less than $18 0^{\circ},$ the endpoints of the angle, $A$ and $B,$ create a minor arc on the circle. This arc can be named $\Arc{AB}.$

If $m ∠ A C B$ is greater than $18 0^{\circ},$ 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,
$∠ A C B$ is a central angle of the circle,
arc $A B$ is a minor arc, and
arc $A D B$ is a major arc.

Rule

Arc Measure

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

Suppose the measure of arc $A B$ is given as $θ .$ The measure of the corresponding major arc, arc $A D B,$ can be found by subtracting $θ$ from $36 0^{\circ},$ the measure of the entire circle.

Concept

Arc Length

Arc length measures the portion of the circle's circumference an arc spans. In the diagram below, the measure of arc $A B$ equals $θ,$ while the length of arc $A B$ 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 $36 0^{\circ} .$

$\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}$

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

\frac{arc length}{2 π r} = \frac{θ}{3 6 0 ^{\circ}}

Find the measure of each arc

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Exercise

Find the measure of arcs $A B$ and $B C A .$

Show Solution

Solution

To begin, let's make sense of the given information. As can be seen,

1.22

is the length of arc

A B,

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.

\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}

To find the measure of arc

A B,

we must first calculate the circumference of the circle.

C C C = 2 π r = 2 π \cdot 2 = 4 π

Now, we can substitute the known values for circumference and arc length into the equation to solve for the measure of arc

A B .

\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}

SubstituteValuesSubstitute values

\frac{1 . 2 2}{4 π} = \frac{arc measure}{3 6 0 ^{\circ}}

UseCalcUse a calculator

0.09708 \dots = \frac{arc measure}{3 6 0 ^{\circ}}

MultEqn

LHS \cdot 36 0^{\circ} = RHS \cdot 36 0^{\circ}

34.9 5^{\circ} \approx arc measure

RearrangeEqnRearrange equation

arc measure \approx 34.9 5^{\circ}

Thus, the measure of arc

A B

is approximately

34.9 5^{\circ} .

Notice that arc

A B

is a minor arc and arc

B C A

is its corresponding major arc. As a result, we can subtract

34.9 5^{\circ}

from

36 0^{\circ}

to find the measure of arc

B C A .

36 0^{\circ} - 34.9 5^{\circ} = 325.0 5^{\circ}

Thus, the measure of arc

B C A

is approximately

325.0 5^{\circ} .

Find the length of the arc

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Exercise

Find the length of arc $A B .$

Show Solution

Solution

To begin, we can make sense of the given information. The circle $C$ has a radius of $4$ inches and the arc $A B$ mesaures $5 0^{\circ} .$ The relationship between arc measure and arc length can be used. $\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}$ First, the circumference of circle $C$ can be calculated as follows. $C C C = 2 π r = 2 π \cdot 4 = 8 π$

The length of arc

A B

can be calculated by substituting the values for circumference and arc measure, and solving.

\frac{arc length}{circumference} = \frac{arc measure}{3 6 0 ^{\circ}}

SubstituteValuesSubstitute values

\frac{arc length}{8 π} = \frac{5 0 ^{\circ}}{3 6 0 ^{\circ}}

CalcQuotCalculate quotient

\frac{arc length}{8 π} = 0.13888 \dots

MultEqn

LHS \cdot 8 π = RHS \cdot 8 π

arc length \approx 3.49

Thus, the length of arc

A B

is approximately

3.49 .

Concept

Radian Measure

In the diagram below, circle $A B$ has a radius of $1$ unit, and circle $C D$ has a radius of $r$ units.

The measures of arcs $A B$ and $C D$ are equal because they are created by the same central angle. Because circle $A B$ has a radius of $1,$ the length of arc $A B$ can be found using the definition of arc length. $arc length A B = \frac{arc measure A B}{3 6 0 ^{\circ}} \cdot 2 π$ Because all circles are similar, circle $A B \sim$ circle $C D .$ Thus, the ratios of corresponding parts between the circles are proportional. An equation relating arc length to radius can be written. $\frac{arc length C D}{arc length A B} = \frac{r}{1}$ Simplifying the right-hand side and substituting the found value for the length of arc $A B$ gives the following. $arc length C D = r \cdot arc length AB = r \cdot (\frac{arc measure A B}{3 6 0 ^{\circ}} \cdot 2 π)$ The equation above shows that the length of arc $C D$ is proportional to the radius of the circle. In fact, the expression $\frac{arc measure A B}{3 6 0 ^{\circ}} \cdot 2 π$

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

2 π,

the radian measure of a circle is

2 π

radians — which corresponds to

36 0^{\circ} .

Rule

Translating between Radians and Degrees

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 $2 π$ radians. It is also known that this same distance can be expressed as $36 0^{\circ} .$ Thus, $2 π radians = 36 0^{\circ} .$ Simplifying gives the following conversion factor.

$π radians = 18 0^{\circ}$

This conversion factor can be used to translate from radians to degrees and vice versa.

Convert between Radians and Degrees

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Exercise

Convert $4 5^{\circ}$ to radians and $\frac{π}{2} radians$ to degrees.

Show Solution

Solution

To convert both measures, we can use the following relation.

π radians = 18 0^{\circ}

The following proportion can be written to convert

4 5^{\circ}

to radians.

\frac{π radians}{1 8 0 ^{\circ}} = \frac{x}{4 5 ^{\circ}}

Solving the equation for

x

will give the corresponding value.

\frac{π radians}{1 8 0 ^{\circ}} = \frac{x}{4 5 ^{\circ}}

MultEqn

LHS \cdot 4 5^{\circ} = RHS \cdot 4 5^{\circ}

\frac{π radians}{1 8 0 ^{\circ}} \cdot 4 5^{\circ} = x

MoveRightFacToNum

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

\frac{π radians \cdot 4 5 ^{\circ}}{1 8 0 ^{\circ}} = x

SimpQuotSimplify quotient

\frac{π radians}{4} = x

UseCalcUse a calculator

0.78539 \dots radians = x

RearrangeEqnRearrange equation

x = 0.78539 \dots radians

Thus,

4 5^{\circ}

corresponds to approximately

0.79

radians. In a similar way, we can convert

\frac{π}{2}

radians to degrees.

\frac{π radians}{1 8 0 ^{\circ}} = \frac{π / 2}{y} \Leftrightarrow y = 9 0^{\circ}

Thus,

\frac{π}{2}

radians corresponds to

9 0^{\circ} .

Rule

Area of a Circle

The area of a circle measures the space inside the circle.

For a circle with the radius $r,$ the area can be calculated using the following formula.

$A = π r^{2}$

Concept

Sector of a Circle

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

In the diagram, sector

A C B

is created by

\overline{A C}, \overline{B C}

and arc

A B .

Rule

Area of a Sector of a Circle

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. $\frac{area of sector}{area of circle} = \frac{measure of arc A B}{measure of circle}$ Substituting the known values gives the following. $\frac{area of sector}{π r ^{2}} = \frac{θ}{3 6 0 ^{\circ}}$ Therefore, the area of a sector of a circle can be found using the following formula.

$area of sector = \frac{θ}{3 6 0 ^{\circ}} \cdot π r^{2}$

Find the area of the sector

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Exercise

Find the area of the sector.

Show Solution

Solution

To begin, notice that the arc corresponding with the given sector measures

10 0^{\circ} .

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.

area of sector = \frac{θ}{3 6 0 ^{\circ}} \cdot π r^{2}

SubstituteII

θ = 10 0^{\circ}

,

r = 12

area of sector = \frac{1 0 0 ^{\circ}}{3 6 0 ^{\circ}} \cdot π \cdot 12^{2}

UseCalcUse a calculator

area of sector \approx 125.66

Thus, the area of the sector is approximately

125.66 cm^{2} .

Using Circumference and Area

Circumference of a Circle

Central Angle

Minor and Major Arc

Arc Measure

Arc Length

Example

Find the measure of each arc

Example

Find the length of the arc

Radian Measure

Translating between Radians and Degrees

Example

Convert between Radians and Degrees

Area of a Circle

Sector of a Circle

Area of a Sector of a Circle

Example

Find the area of the sector