Topics
11. Circles
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United States of America
Grades 9-12

11.01 Radians, arc length, and sector area

Lesson
Practice

Introduction

When we consider part of a circle in relation to the whole circle, the measurements of that part of the circle are easy to find. These proportional relationships lead to several other interesting properties that will be explored in this lesson.

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.

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

Examples

Example 1

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

a

Find the length of the solid arc.

Worked Solution
Create a strategy

The full circle is 360\degree and has a length of 144\text{ 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\text{ cm}

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

b

Describe the relationship of the arc length to the circumference for any circle, and develop a formula for the arc length of any sector.

Worked Solution
Create a strategy

Recall the circumference of a circle is C=2\pi r. Let \theta represent the central angle of a circle and s represent the arc length created by the radii of the central angle.

Apply the idea

The central angle of a sector is a proportion of the full rotation of the circle. The arc length will be the same proportion of the circle's circumference

\displaystyle s\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot C
\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot 2\pi r
Reflect and check

To solve for the arc length of any circle, we need to know central angle and either the circumference, radius or diameter.

Example 2

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

Worked Solution
Create a strategy

In the previous example, we found the formula for the arc length of a sector to be s=\dfrac{\theta}{360}\cdot 2\pi r. We can use this formula with the values of the arc length and radius given in the diagram to solve for the central angle, \theta.

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 is about 171.89\degree.

Example 3

For the following sector, where AB = 5 inches:

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

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 5\text{ in}:

\displaystyle A_{\text{circle}}\displaystyle =\displaystyle \pi (5)^2
\displaystyle =\displaystyle 25\pi\text{ 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}\, \text{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.

b

Describe the relationship of the sector area to the area of the circle, and generalize a formula for finding the sector area in any circle.

Worked Solution
Create a strategy

We are describing the area of the sector in relation to the area of the full circle. Because there are two variables for area, we need to distinguish between them using subscripts.

Apply the idea

The central angle of the sector is a proportion of a full rotation of the circle. The sector area is the same proportion of the full circle's area.

\displaystyle A_{\text{sector}}\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot A_{\text{circle}}
\displaystyle =\displaystyle \dfrac{\theta}{360}\cdot \pi r^2

Example 4

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

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\text{ m} and angle 90\degree.

In the previous example, we found the formula for the arc length of a sector to be 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^2Area of a sector formula
\displaystyle =\displaystyle \dfrac{90}{360}\cdot \pi \left(9\right)^2Substitute 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\text{ 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 = \frac{\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 = \frac{\theta}{360} \pi r^2
\bm{A}
Area of the sector
\bm{\theta}
Central angle
\bm{r}
Radius

Circle similarity

The full length around a circle is known as its circumference. If we rewrite the formula for the circumference of a circle, we see \dfrac{C}{d}=\pi. This means that the ratio between the circumference and diameter of every circle is equal to the constant \pi.

Since corresponding parts of similar figures are proportional, and \dfrac{C}{d} = \pi for a circle of any diameter, all circles are similar.

Exploration

Circle O\rq has been dilated by a scale factor of 2.

Circle O
Circle O\rq

  1. Compare each of the following parts of circle O\rq to the corresponding parts of circle O.

  2. Determine the factor by which each part of circle O\rq has changed.

    • Radius
    • Central angle
    • Arc length
    • Ratio of the circle's circumference to its diameter

Because all circles are similar, their corresponding parts are proportional to each other.

Examples

Example 5

Determine a sequence of transformations to map circle C onto circle C\rq.

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Worked Solution
Create a strategy

To map circle C onto circle C\rq, we will need to translate it so that the circles are concentric, meaning they have the same center. We can also observe that circle C is larger than circle C\rq which means that we will need to dilate circle C by a scale factor that is less than 1.

The axes are labeled by twos, so we need to be careful to correctly count the units when translating and dilating.

Apply the idea

First, we will translate circle C to make it concentric with circle C\rq. To do this, we need to translate circle C right 10 units and up 5 units. The transformation mapping that represents this is \left(x,y\right)\to \left(x+10, y+5\right).

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Next, we need to determine the factor by which we must dilate circle C. By counting, we find the radius of circle C to be 6 and the radius of circle C\rq to be 3. If we let the scale factor be k,

\displaystyle 6\cdot k\displaystyle =\displaystyle 3
\displaystyle k\displaystyle =\displaystyle \dfrac{1}{2}

Circle C should be dilated by a factor of k=\dfrac{1}{2} with the center of dilation being the center of the circles at \left(6,1\right).

Therefore, translating circle C right 10 units, up 5 units, then dilating by a scale factor of \dfrac{1}{2} will map circle C onto circle C\rq.

Reflect and check

We could have dilated circle C first using the origin as the center of dilation and a scale factor of k=\dfrac{1}{2}.

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Then, translate circle C right 8 units and up 3 units. Either way, we are still only using a translation and a dilation to map circle C onto circle C\rq.

Example 6

Prove all circles are similar using transformation mapping.

Worked Solution
Create a strategy

As we discussed in  7.02 Similarity transformations  , any combination of rotations, translations, reflections, and dilations will maintain similarity. We can prove all circles similar if we can find a combination of the above transformations that maps any circle onto another.

We will begin by drawing two circles of different sizes and labeling their radii.

Apply the idea

We can translate circle A onto circle B by a vector \overrightarrow{AB}.

Now, the circles are concentric which makes it easier to determine the next transformation needed.

From the image, we can see that we need to dilate circle A (the smaller circle). To dilate it, we need to multiply the radius by a constant scale factor, k, to make it equal to the radius of circle B.k\cdot r_1=r_2 where k=\dfrac{r_2}{r_1}.

Since any two circles can be mapped by a sequence of a translation and a dilation by a scale factor of \dfrac{r_2}{r_1}, all circles are similar.

Example 7

Prove that any two sectors with the same central angle will have an arc length that is proportional to its radius.

Worked Solution
Create a strategy

Since all circles are similar, we can use the corresponding parts of similar figures to prove this statement.

Apply the idea

Corresponding parts of similar figures are proportional to each other. This means the radii and arc lengths are dilated by the same constant of proportionality. From this, we know \dfrac{r_1}{r_2}=\dfrac{s_1}{s_2}

If we rewrite this to solve for the arc length of the smaller figure, we get s_1=\left(\dfrac{s_2}{r_2}\right)r_1

This shows that the arc length of a circle is directly proportional to its radius. The constant of proportionality is \dfrac{s_2}{r_2}.

Idea summary

We can prove all circles are similar by using similarity transformations to map one circle onto another. Because all circles are similar, their corresponding parts are proportional. The arc length of a circle's central angle is directly proportional to its 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.

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 8

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 9

Convert the following radians to degrees.

a

1.8\text{ 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\text{ 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}\text{ 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}\text{ 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}\text{ rad}=120\degree.

Example 10

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\text{ 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 11

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

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.C.A.1

Prove that all circles are similar.

G.C.B.5

Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector.

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