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.

Concept

Circle

A circle is the set of all points that are equidistant from a given point. The given point is usually called the center of the circle. The distance between the center and any point on the circle is called a radius.

Concept

Circumference of a Circle

The circumference is the distance around a circle, otherwise known as the circle's perimeter.

The circumference is calculated by multiplying the diameter of the circle by

π .

$C = π d$

Since the diameter is twice the radius, it can also be calculated by multiplying $2 r$ by $π .$

$C = 2 π r$

Concept

Central Angle

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.

Why

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.

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

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

In the diagram, the measure of $\Arc{AB}$ equals

θ,

while the length of $\Arc{AB}$ 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 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.

$A = π r^{2}$

Proof

Start by dividing a circle with radius $r$ into a number of equally sized sectors. Then distinguish the top and bottom half of the circle by filling them with different colors. Because the circumference of a circle is $2 π r,$ the arc length of each semicircle is half of it, $π r .$

Now, imagine that all of the sectors of the circle are unfold. By placing the blue part as teeth pointing downward and the green part 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.

It is not easy to calculate the area of this figure, but note that if the circle is divided into more, even smaller sectors, then the figure will begin to look like a rectangle more and more.

The vertical sides becomes more vertical, and the horizontal sides becomes more horizontal. If the circle is divided into infinitely small sectors, the figure will become a perfect rectangle, with base $π r$ and height $r .$ Since the area of a rectangle is the product of its height and its base, the area of the figure created by the sectors of circle can be found as follows. $A = π r \cdot r = π r^{2}$ Therefore, the area of the circle can also be found by this formula.

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

Circle

Circumference of a Circle

Central Angle

Why

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

Proof

Sector of a Circle

Area of a Sector of a Circle

Example

Find the area of the sector