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12. Circles
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Geometry

12.01 Arc length and sector area

Lesson
Practice

Arc length and sector area

Recall the circumference of a circle is the distance around the edge of the circle. It is calculated by C = 2 \pi r. A part of the circumference of a circle is called an arc. The distance from one endpoint of the arc to the other endpoint is called the arc length. If one endpoint is A and the other is B, we denote the arc length by \overset{\large\frown}{AB}.

Any arc of a circle has a corresponding central angle, and together, the arc and central angle form a sector. We can use the ratio between the central angle or arc length of the sector to the whole circle to find the area of the sector.

Central angle

An angle that has its vertex at the center of a circle with radii as its sides.

A circle with two radii drawn. The angle between them is marked
Sector

A region inside a circle bounded by an arc and the two radii which form its central angle.

A circle with two radii. The smaller area of the circle between the two radii is highlighted

Exploration

Drag the sliders to change the attributes of the circle.

Loading interactive...
  1. As you change the radius, what else changes? What stays the same?
  2. As you change the measure of the central angle, what else changes? What stays the same?
  3. What do you notice about the ratios?
  4. Do you think there is a similar relationship between the area of the sector and the area of the whole circle? Explain.

The arc length and area of a sector in a circle are both proportional to the central angle of the sector. The arc length measures the distance along the circle's edge for the sector, while the area of the sector measures the region inside the circle that the sector covers.

For example, if a sector has a central angle of 90 \degree, it takes up one-fourth of the circle's full angle\left(360 \degree \right), meaning its arc length will be one-fourth of the entire circumference, and its area will be one-fourth of the entire area of the circle.

Two circles, one small and one large. Both have a central angle of 90 degrees. The small circle has a radius labeled 5 centimeters and the large circle has a radius labeled 10 centimeters.

The radius also affects these values: if the radius doubles, both the arc length and area increase, but the area increases by a factor of four (since area depends on the square of the radius), while arc length only doubles.

We can calculate arc length as a proportion of the total circumference by considering the central angle of the arc as a proportion of a full rotation:

\displaystyle s = \dfrac{\theta}{360} \cdot 2 \pi r
\bm{s}
Arc length
\bm{\theta}
Central angle
\bm{r}
Radius

We can calculate the area of the sector in a similar way to its arc length, by taking a proportion of the total area of the circle corresponding to the central arc's proportion of a full rotation:

\displaystyle A = \dfrac{\theta}{360} \pi r^2
\bm{A}
Area of the sector
\bm{\theta}
Central angle
\bm{r}
Radius

Examples

Example 1

The circumference of the given circle is 144 \operatorname{cm}.

A sector with an angle of 160 degrees. Ask your teacher for more information.
a

Determine the ratio of the central angle to the total number of degrees in the circle.

Worked Solution
Create a strategy

The total number of degrees in a circle is 360 \degree. The ratio is written as \dfrac{\text{central angle}}{360} and simplified.

Apply the idea

The measure of the central angle is 160 \degree.

\displaystyle \dfrac{\text{central angle}}{360}\displaystyle =\displaystyle \dfrac{160}{360}
\displaystyle \dfrac{160 \div 40}{360 \div 40}\displaystyle =\displaystyle \dfrac{4}{9}Reduce by dividing by 40

The simplified ratio is \dfrac{4}{9}.

Reflect and check

The ratio of the central angle to the total number of degrees in a circle is proportional to the ratios of other parts of a circle. For example, the ratio of the arc length to circumference and the sector area to the area of the circle are both proportional to the ratio of the central angle to 360 \degree.

b

Find the length of the solid arc.

Worked Solution
Create a strategy

The full circle is 360\degree and has a length of 144 \operatorname{cm}. If we can determine what proportion 160 \degree is of the circle, then we can use that to determine the length of the arc of that sector as a proportion of the circumference.

Apply the idea

The angle in the middle is \dfrac{160}{360}=\dfrac{4}{9} of the entire circle, so the arc length of the sector is going to be \dfrac{4}{9} of the circumference.

\displaystyle \dfrac{4}{9}\cdot C_{\text{circle}}\displaystyle =\displaystyle \dfrac{4}{9}\cdot 144
\displaystyle =\displaystyle 64\operatorname{ cm}

The arc length of the sector is 64 \operatorname{ cm}.

Example 2

The sector shown has a radius of 10 \operatorname{cm} and an arc length of 30 \operatorname{cm}. Find the measure of the central angle \theta in degrees. Round your answer to two decimal places.

A sector with a radius of 10 centimeters, arc length of 30 centimeters, and center angle of theta.
Worked Solution
Create a strategy

Use the formula for the arc length of a sector: s=\dfrac{\theta}{360}\cdot 2\pi r.

Apply the idea

We were given s=30 and r=10.

\displaystyle s\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot 2\pi rFormula for arc length
\displaystyle 30\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot 2\pi \cdot 10Substitute known values
\displaystyle 30\displaystyle =\displaystyle \dfrac{\theta \cdot \pi}{18}Evaluate the multiplication and division
\displaystyle 540\displaystyle =\displaystyle \theta \cdot \piMultiply by 18
\displaystyle 171.89\degree\displaystyle \approx\displaystyle \thetaDivide by \pi

The central angle of the sector measures approximately 171.89\degree.

Example 3

For the sector shown, AB = 5 inches:

Sector B C of circle A. Central angle B A C has a measure of 68 degrees.

Find the area of the sector.

Worked Solution
Create a strategy

We can find the area of the sector using a process similar to finding the arc length of the sector. First, we determine the central angle's proportion of the full circle. The sector's area is the same proportion of the full circle's area.

Apply the idea

Recall the area of a circle is A=\pi r^2. Let's begin by finding the area of the full circle with radius 15 \operatorname{in}:

\displaystyle A_{\text{circle}}\displaystyle =\displaystyle \pi (5)^{2}
\displaystyle =\displaystyle 25\pi\operatorname{in}^{2}

Next, we need to determine the central angle's proportion of the full circle: \dfrac{68}{360}=\dfrac{17}{90} This means the area of the sector will be \dfrac{17}{90}ths of the area of the circle.

\displaystyle A_{\text{sector}}\displaystyle =\displaystyle \dfrac{17}{90}\cdot A_{\text{circle}}
\displaystyle =\displaystyle \dfrac{17}{90}\cdot 25\pi
\displaystyle =\displaystyle \dfrac{85\pi}{18}

The area of the sector is \dfrac{85 \pi }{18}\, \operatorname{in}^{2}.

Reflect and check

Since there was no instruction to approximate the solution by rounding the answer, we should keep the answer as an exact value.

Example 4

A goat is tethered to a corner of a fenced field as shown. The fences are perpendicular. The rope is 9 \operatorname{m} long. Find the area of the field the goat can graze over. Give the answer correct to two decimal places.

The image shows a goat is tethered to a corner of a fenced field.
Worked Solution
Create a strategy

Notice that the angle at the corner of the fence is 90\degree. The area that the goat can graze over is a sector of radius 9 \operatorname{m} and angle 90\degree.

Use the formula for the arc length of a sector: A=\dfrac{\theta}{360}\cdot \pi r^2. We can substitute the values of the central angle and radius into the formula to solve for the sector area.

Apply the idea

We were given \theta=90 and r=9.

\displaystyle A\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot \pi r^{2}Area of a sector formula
\displaystyle =\displaystyle \dfrac{90}{360}\cdot \pi \left(9\right)^{2}Substitute known values
\displaystyle =\displaystyle \dfrac{81\pi}{4}Evaluate the multiplication and division
\displaystyle \approx\displaystyle 63.62Approximate to two decimal places

The area of the field the goat can graze over is about 63.62\operatorname{ m}^{2}.

Idea summary

The distance from one endpoint, A, of an arc to the other endpoint, B, is called the arc length and is denoted as \overset{\large\frown}{AB}. We can calculate this as a proportion of the total circumference by considering the central angle of the arc as a proportion of a full rotation:

\displaystyle s = \dfrac{\theta}{360} \cdot 2 \pi r
\bm{s}
Arc length
\bm{\theta}
Central angle
\bm{r}
Radius

We can calculate the area of the sector in a similar way to its arc length, by taking a proportion of the total area of the circle corresponding to the central arc's proportion of a full rotation:

\displaystyle A = \dfrac{\theta}{360} \pi r^2
\bm{A}
Area of the sector
\bm{\theta}
Central angle
\bm{r}
Radius

Degrees and radians

We have established that sectors with the same central angle will have an arc length that is proportional to the radius. We define this constant of proportionality as the radian measure of the central angle.

Radian measure (of a central angle)

The ratio of the arc length divided by the radius, \theta=\dfrac{s}{r}, where \theta is the central angle in radians, s is the arc length, and r is the radius of the circle.

From the time of the ancient Babylonians, it has been the practice to divide circles into 360 small arcs. The central angle of any one of those arcs is called one degree. In effect, an arc of the circle is used as a measure of its central angle.

We have defined the central angle in radians as the ratio of the arc length divided by the radius. So, the central angle of an arc whose length is equal to the radius is 1 radian.

A circle with a radius labeled r. Inside the circle, a sector is highlighted that represents an angle of 1 radian. The sector is defined by two radii drawn from the center to the circumference of the circle, and the intercepted arc of the circle. The arc and the two radii create a shape that looks like a slice of pie or a triangle with a curved base. The ends of the arc are connected by a straight line that represents the length of the arc, which is also the length of the radius, indicating the definition of a radian.

Radians are an alternate way to describe angles and are the international standard unit for measuring angles. Because angles in radian measure are in essence just fractions of the circle, they do not require a unit.

Exploration

Drag the slider to change the radius. Move the point to change the size of the angle.

Loading interactive...

Use the applet to answer the following questions:

  1. What do you notice about a sector with a radian measure of 1?

  2. What is the measure of 1 radian in degrees?

  3. What is the measure of the central angle of a semicircle in degrees and radians?

  4. What is the measure of the central angle of a full circle in degrees and radians?

  5. How can we use this information to convert from radians to degrees and from degrees to radians?

As we previously explored in this lesson, the arc length of a sector is part of the circumference of a full circle. We just learned that a radian is defined as the ratio of the arc length and the radius. This means the central angle in radians of a full circle is \theta=\dfrac{2\pi r}{r}=2\pi When we compare this to degrees, we see that: \begin{aligned}2\pi\text{ rad}&=360\degree\\\pi\text{ rad}&=180\degree\end{aligned}

Examples

Example 5

Convert the following degrees to radians.

a

90\degree

Worked Solution
Create a strategy

We can use proportions to find this result since we know 180\degree=\pi radians. First, we will need to find what proportion 90\degree is of 180\degree.

Apply the idea

90\degree is \dfrac{1}{2} of 180\degree, so in radians, the measure will be \dfrac{1}{2} of \pi.\dfrac{1}{2}\cdot \pi = \dfrac{\pi}{2}

This shows 90\degree=\dfrac{\pi}{2} radians.

Reflect and check

We could also have written the proportion as \dfrac{90}{180}=\dfrac{x}{\pi} Solving for x, we get \dfrac{90}{180}\pi=x Notice that this is the same as multiplying the angle by \dfrac{\pi}{180}.

90\cdot \dfrac{\pi}{180}= \dfrac{\pi}{2}

b

216\degree

Worked Solution
Create a strategy

As we found in the reflection of the previous part, we can multiply an angle in degrees by \dfrac{\pi}{180} to convert it to radians.

Apply the idea

216\cdot\dfrac{\pi}{180} =\dfrac{6\pi}{5}

This shows 216\degree=\dfrac{6\pi}{5} radians.

Reflect and check

Using proportions to check our answer:\dfrac{216}{360}=\dfrac{3}{5}

216\degree is \dfrac{3}{5} of the whole circle. In radians, the full rotation of the circle is 2\pi.

\dfrac{3}{5}\cdot 2\pi =\dfrac{6\pi}{5}

\dfrac{3}{5} of the circle in radians is \dfrac{6\pi}{5}.

Example 6

Convert the following radians to degrees.

a

1.8\operatorname{ rad}

Round to one decimal place.

Worked Solution
Create a strategy

Since we know \pi=180\degree, we will set up an equivalent proportion between radians and degrees and solve for the angle in degrees.

Apply the idea
\displaystyle \dfrac{1.8}{\pi}\displaystyle =\displaystyle \dfrac{x}{180}
\displaystyle \dfrac{1.8}{\pi}\cdot 180\displaystyle =\displaystyle x

Using a calculator to evaluate, then rounding to one decimal place, we find 1.8\operatorname{ rad}\approx 103.1\degree.

Reflect and check

When we set up an equivalent proportion, we converted from radians to degrees by multiplying the radian measure by \dfrac{180}{\pi}.

b

\dfrac{2\pi}{3}\operatorname{ rad}

Worked Solution
Create a strategy

As we found in the reflection of the previous part, we can multiply an angle in radians by \dfrac{180}{\pi} to convert it to degrees.

Apply the idea

\dfrac{2\pi}{3}\cdot\dfrac{180}{\pi}=120\degree

This shows \dfrac{2\pi}{3}\operatorname{ rad}=120\degree.

Reflect and check

We can check this answer by finding what part of the circle \dfrac{2\pi}{3} radians is, then determine if 120\degree is the same fraction of the circle in degrees.

\displaystyle \dfrac{\frac{2\pi}{3}}{2\pi}\displaystyle =\displaystyle \dfrac{1}{3}
\displaystyle \dfrac{120}{360}\displaystyle =\displaystyle \dfrac{1}{3}

Both are \dfrac{1}{3} of the full circle, so this shows \dfrac{2\pi}{3}\operatorname{ rad}=120\degree.

Example 7

The sector of a circle with radius 7 is formed from an angle of size \dfrac{5\pi}{4}.

a

Find the exact length of the arc.

Worked Solution
Create a strategy

We defined the measure of a radian as the ratio of the arc length to the radius, \theta = \frac{s}{r}. We can use this formula to solve for the arc length.

Apply the idea

We were given \theta=\dfrac{5\pi}{4} and r=7.

\displaystyle \theta\displaystyle =\displaystyle \dfrac{s}{r}Definition of a radian
\displaystyle \dfrac{5\pi}{4}\displaystyle =\displaystyle \dfrac{s}{7}Substitute known values
\displaystyle \dfrac{35\pi}{4}\displaystyle =\displaystyle sMultiply both sides by 7

The arc length is \dfrac{35\pi}{4} units.

Reflect and check

By the definition of a radian, we can find the length of any arc when given the radius and central angle in radians by s = r\cdot \theta.

b

Find the area of the sector.

Worked Solution
Create a strategy

We derived the sector area formula in degrees by taking the amount of the circle covered by the sector and multiplying by the total area of the circle. We can do the same thing in radians since 360\degree=2\pi\operatorname{ rad}.

\displaystyle A\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot \pi r^2Sector area formula in degrees
\displaystyle A\displaystyle =\displaystyle \dfrac{\theta}{2\pi}\cdot \pi r^2Sector area formula in radians
\displaystyle =\displaystyle \dfrac{\theta\cdot r^2}{2}Simplify since \dfrac{\pi}{\pi}=1
Apply the idea

Using the formula for the area of a sector in radians with \theta=\dfrac{5\pi}{4} and r=7:

\displaystyle A\displaystyle =\displaystyle \dfrac{\theta\cdot r^{2}}{2}Sector area formula in radians
\displaystyle =\displaystyle \dfrac{\frac{5\pi}{4}\cdot\left(7\right)^{2}}{2}Substitute known values
\displaystyle =\displaystyle \dfrac{\frac{245\pi}{4}}{2}Evaluate the numerator
\displaystyle =\displaystyle \dfrac{245\pi}{4}\cdot \dfrac{1}{2}Evaluate the division
\displaystyle =\displaystyle \dfrac{245\pi}{8}Evaluate the multiplication

The area of the sector is \dfrac{245\pi}{8}\text{ units}^{2}.

Example 8

The sector below has an area of 3.6\operatorname{m}^{2} and a radius of 2\operatorname{m}. Find the value of \theta in the sector below.

a sector of a circle with a central angle denoted by theta. The radius of the sector is labeled as 2 meters. The arc of the sector is curved, and the two radii extend from the center of the circle to the ends of the arc, creating a pie-slice shape.
Worked Solution
Create a strategy

We are given the area and the radius of the sector so we can use the formula A=\dfrac{\theta\cdot r^{2}}{2} to solve for \theta.

Apply the idea
\displaystyle A\displaystyle =\displaystyle \dfrac{\theta\cdot r^2}{2}Sector area formula in radians
\displaystyle 3.6\displaystyle =\displaystyle \dfrac{\theta\cdot 2^{2}}{2}Substitute A=3.6 and r=2
\displaystyle 7.2\displaystyle =\displaystyle \theta\cdot 4Multiply by 2 and evaluate the exponent
\displaystyle 1.8\displaystyle =\displaystyle \thetaDivide by 4

Therefore, the central angle of the sector is 1.8 radians.

Idea summary

A radian measure of the central angle of a sector is defined as the ratio of the arc length divided by the radius of the sector, \theta = \dfrac{s}{r}.

To convert from degrees to radians, multiply by \dfrac{\pi}{180}

To convert from radians to degrees, multiply by \dfrac{180}{\pi}

The formula for the arc length of a sector in radians is

\displaystyle s=\theta\cdot r
\bm{\theta}
Central angle measure in radians
\bm{r}
Radius

The formula for the area of a sector in radians is

\displaystyle A=\dfrac{\theta\cdot r^{2}}{2}
\bm{\theta}
Central angle measure in radians
\bm{r}
Radius

Outcomes

G.PC.3

The student will solve problems, including those in context, by applying properties of circles.

G.PC.3a

Determine the proportional relationship between the arc length or area of a sector and other parts of a circle.

G.PC.3d

Calculate the length of an arc of a circle.

G.PC.3e

Calculate the area of a sector of a circle.

G.PC.3f

Apply arc length or sector area to solve for an unknown measurement of the circle including the radius, diameter, arc measure, central angle, arc length, or sector area.

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