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 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 $C .$

In the applet, the center is labeled

O .

Therefore, the circle can be referred to as

⊙ O,

or circle

O .

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

Rule

Circumference of a Circle

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

$C = π d$

This can be visualized in the following diagram.

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

$C = 2 π r$

Proof

Consider two circles and their respective diameters and circumferences.

By the Similar Circles Theorem, all circles are similar. Therefore, their corresponding parts are proportional.

\frac{C _{B}}{C _{A}} = \frac{d _{B}}{d _{A}}

This proportion can be rearranged.

\frac{C _{B}}{C _{A}} = \frac{d _{B}}{d _{A}}

Rearrange equation

MultEqn

$LHS \cdot C_{A} = RHS \cdot C_{A}$

C_{B} = \frac{d _{B}}{d _{A}} (C_{A})

DivEqn

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

\frac{C _{B}}{d _{B}} = \frac{1}{d _{A}} (C_{A})

MoveRightFacToNumOne

$\frac{1}{b} \cdot a = \frac{a}{b}$

\frac{C _{B}}{d _{B}} = \frac{C _{A}}{d _{A}}

Therefore, for any two circles, the ratio of the circumference to the diameter is always the same. This means that this ratio is constant. This constant is defined as

π .

With this information, it can be shown that the circumference of a circle is the product between its diameter and

π .

$\frac{C}{d} = π \Rightarrow C = π d$

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 $A B .$

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 $A D B .$ 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.

Concept

Arc Measure

The measure of an arc is equal to the measure of its corresponding central angle. In the diagram, the measure of $A B$ is equal to the measure of $∠ A C B .$

Suppose the measure of $A B$ is given as $θ .$ The measure of the corresponding major 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

A B

equals

θ,

while the length of

A B

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

fullscreen

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}}

SubstituteValues

Substitute values

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

UseCalc

Use 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

RearrangeEqn

Rearrange 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

fullscreen

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}}

SubstituteValues

Substitute values

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

CalcQuot

Calculate 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

fullscreen

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

SimpQuot

Simplify quotient

\frac{π radians}{4} = x

UseCalc

Use a calculator

0.78539 \dots radians = x

RearrangeEqn

Rearrange 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 is the product of $π$ and the square of its radius.

Proof

Informal Justification

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 $2 π r,$ the arc length of each semicircle is half this value, $π r .$

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.

It can be noted that if the circle is divided into more and smaller sectors, then the figure will begin to look more and more like a rectangle.

Here, the shorter sides become more vertical and the longer sides become more horizontal. If the circle is divided into infinitely many 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 following formula can be derived.

$A = π r \cdot r \Leftrightarrow A = π r^{2}$

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

Concept

Sector of a Circle

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

In the diagram, sector

A C B

is created by

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

and

A B .

Rule

Area of a Sector of a Circle

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 $36 0^{\circ} .$

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

From the fact that $2 π rad$ equals $36 0^{\circ},$ an equivalent formula can be written if the central angle is given in radians.

$Area of Sector Area of Sector = \frac{θ}{2 π} \cdot π r^{2} ⇓ = \frac{θ}{2} \cdot r^{2}$

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

$Area of Sector = \frac{m A B}{3 6 0 ^{\circ}} \cdot π r^{2}$
or
$Area of Sector = \frac{m A B}{2} \cdot r^{2}$

Proof

Consider sector $A C B$ bounded by $\overline{A C}, \overline{B C},$ and $A B .$

Since a circle measures $36 0^{\circ},$ this sector represents $\frac{θ}{3 6 0 ^{\circ}}$ of $⊙ C .$ Therefore, the ratio of the area of a sector to the area of the whole circle is proportional to $\frac{θ}{3 6 0 ^{\circ}} .$ $\frac{Area of Sector}{Area of Circle} = \frac{θ}{3 6 0 ^{\circ}}$

Recall that the area of a circle is $π r^{2} .$ By substituting it into the equation and solving for the area of a sector, the desired formula can be obtained. $\frac{Area of Sector}{π r ^{2}} = \frac{θ}{3 6 0 ^{\circ}}$ Therefore, the area of a sector of a circle can be found by using the following formula.

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

Find the area of the sector

fullscreen

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}

UseCalc

Use a calculator

area of sector \approx 125.66

Thus, the area of the sector is approximately

125.66 cm^{2} .

Using Circumference and Area

Concept

Circle

Rule

Circumference of a Circle

Proof

Concept

Central Angle

Why

Concept

Minor and Major Arc

Concept

Arc Measure

Concept

Arc Length

Example

Find the measure of each arc

Example

Find the length of the arc

Concept

Radian Measure

Rule

Translating between Radians and Degrees

Example

Convert between Radians and Degrees

Rule

Area of a Circle

Proof

Concept

Sector of a Circle

Rule

Area of a Sector of a Circle

Proof

Example

Find the area of the sector