Find arc length, sector area, and radian measure

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Difficulty: Intermediate
Estimated time: 28 minutes
Domains: Geometry and Trigonometry
Techniques: Arc-lengthSector-areaFraction-of-circleDegree-radian-conversionAngle-positioning

What you’ll learn

Turn a central angle into a fraction of the whole circle.
Tell an arc’s degree measure apart from its actual length.
Find arc length and sector area, with the angle in degrees or radians.
Work backward to the angle when you know an arc length or a sector’s area.
Convert exactly between degrees and radians.
Place common radian angles in standard position (finding their sine and cosine comes later).

Prerequisites

Why this matters on the SAT

Turn the central angle into a fraction of the circle

Arc and sector questions can hand you a central angle, a radius, the circumference, an arc length, or a sector’s area. Whatever they give you, the key question is the same: what fraction of the whole circle does the angle take up? Here’s a typical one.

Solution to the example

A full turn is $360^\circ$ , so the $80^\circ$ angle takes up

\frac{80}{360}=\frac29

of the circle. The arc gets that same fraction of the circumference:

\text{arc length}=\frac29(45)=10.

The answer is C. Notice that the same arc has two different numbers: it measures $80^\circ$ , but it’s $10$ units long.

SAT example

The

80^\circ

angle selects the same fraction of the circumference and the circle’s area.

A circle has circumference $45$ units. A central angle measuring $80^\circ$ intercepts minor arc $\widehat{AB}$ . What is the length, in units, of minor arc $\widehat{AB}$ ?

A
$8$
B
$9$
C
$10$
D
$12$

Use one fraction for arcs and sectors

A sector is a slice of a circle, like a slice of pizza: the region between two radii and the arc they cut off. If its central angle is $\theta^\circ$ , the sector takes up

\boxed{\frac{\theta}{360}}

of the circle.

That one fraction works on two different wholes:

\boxed{\text{arc length} =\frac{\theta}{360}(2\pi r)}

and

\boxed{\text{sector area} =\frac{\theta}{360}(\pi r^2)}.

Arc length is a length, so it comes in ordinary units, like centimeters. Sector area is an area, so it comes in square units, like square centimeters.

For example, a $90^\circ$ angle takes up $\frac{90}{360}=\frac14$ of a circle. Its arc is a quarter of the circumference, and its sector is a quarter of the area.

Check your understanding:

What fraction of a circle does a $144^\circ$ central angle take up? If the circumference is $35\pi$ centimeters, how long is its arc?

Common mistake:

Since a central angle and its arc have the same degree measure, it’s tempting to call an $80^\circ$ arc $80$ units long. But the arc’s length also depends on how big the circle is: the same $80^\circ$ arc is longer on a bigger circle. Find the fraction of the circle, then multiply it by the circumference.

The fraction also runs backward. When you know an arc or a sector and need the angle, set the part over its whole equal to the angle over $360^\circ$ . For an arc of length $s$ ,

\frac{s}{2\pi r}=\frac{\theta}{360}.

For a sector of area $K$ ,

\frac{K}{\pi r^2}=\frac{\theta}{360}.

It’s the same idea both ways: the part is the same fraction of its whole as the angle is of $360^\circ$ .

Convert degrees and radians

Degrees and radians are two units for the same thing, the size of a turn, much like inches and centimeters are two units for length. One full turn is $360^\circ$ , and it’s also $2\pi$ radians. Halve both and you get the fact to remember:

\boxed{180^\circ=\pi\text{ radians}}.

Use it like any unit conversion:

To go from degrees to radians, multiply by $\frac{\pi}{180^\circ}$ .
To go from radians to degrees, multiply by $\frac{180^\circ}{\pi}$ .

For $135^\circ$ ,

135^\circ\left(\frac{\pi}{180^\circ}\right) =\frac{3\pi}{4}.

The degrees cancel, so you’re left with radians.

For $\frac{7\pi}{6}$ radians,

\frac{7\pi}{6}\left(\frac{180^\circ}{\pi}\right) =210^\circ.

This time the $\pi$ cancels, so you’re left with degrees.

Check your understanding:

Convert $300^\circ$ to radians and $\frac{5\pi}{12}$ radians to degrees.

Common mistake:

It’s easy to flip the two factors. To pick the right one every time, write the unit you’re starting with beside the angle. Then choose the factor that cancels that unit and leaves the one you want.

Position a radian angle

An angle is in standard position when it starts on the positive $x$ -axis. A positive angle turns counterclockwise from there, the opposite way from a clock’s hands.

The quarter turns are landmarks for placing the ending ray.

Each quarter turn lands on an axis, and those four spots are your landmarks:

\begin{array}{c|c} \text{Where the angle ends} & \text{Radian measure}\\[0.9em] \hline \text{positive }x\text{-axis} & 0\text{ or }2\pi\\[0.9em] \text{positive }y\text{-axis} & \frac{\pi}{2}\\[0.9em] \text{negative }x\text{-axis} & \pi\\[0.9em] \text{negative }y\text{-axis} & \frac{3\pi}{2} \end{array}

To place $\frac{2\pi}{3}$ , find the two landmarks it sits between:

\frac{\pi}{2}<\frac{2\pi}{3}<\pi.

It’s more than a quarter turn but less than a half turn, so its ending ray lands between the positive $y$ -axis and the negative $x$ -axis, in quadrant II. Converting gives $120^\circ$ , which lands in the same place.

Check your understanding:

In which quadrant does $\frac{5\pi}{4}$ end? Use the landmarks to explain.

Use radians directly in circle formulas

In radians, a full turn is $2\pi$ instead of $360$ , so an angle $\theta$ takes up

\frac{\theta}{2\pi}

of the circle. Put that fraction into the arc-length formula, and the $2\pi$ cancels:

\begin{aligned} \text{arc length} &=\frac{\theta}{2\pi}(2\pi r)\\[1.4em] &=\boxed{r\theta}. \end{aligned}

Sector area shrinks the same way:

\begin{aligned} \text{sector area} &=\frac{\theta}{2\pi}(\pi r^2)\\[1.4em] &=\boxed{\frac12r^2\theta}. \end{aligned}

These shortcuts need $\theta$ in radians.

Take a circle of radius $5$ and a central angle of $\frac{3\pi}{5}$ radians:

s=r\theta =5\left(\frac{3\pi}{5}\right) =3\pi.

The fraction gives the same answer: $\frac{3\pi/5}{2\pi}=\frac3{10}$ , and $\frac3{10}$ of the circumference $10\pi$ is $3\pi$ .

Common mistake:

If the angle has a degree symbol, $s=r\theta$ goes badly wrong. With $60^\circ$ it would make the arc $60r$ long, nearly ten times the whole circumference, $2\pi r$ . For degrees, use the fraction $\frac{\theta}{360}$ , or convert the angle to radians first.

Example: Find a sector area

Worked example

A circle has radius $12$ centimeters. A sector has a central angle of $150^\circ$ . Which choice gives the area, in square centimeters, of the sector?

A
$24\pi$
B
$50\pi$
C
$60\pi$
D
$144\pi$

Step 1

Name the whole and the part

The question asks for a sector’s area, so the whole is the circle’s area:

A_{\text{circle}}=\pi r^2=\pi(12)^2=144\pi.

Step 2

Find the fraction of the circle

The sector takes up

\frac{150}{360}=\frac5{12}

of the circle.

Step 3

Take that fraction of the area

Cancel the $12$ into $144$ before you multiply, and the numbers stay small:

\begin{aligned} A_{\text{sector}} &=\frac5{12}(144\pi)\\[1.4em] &=5(12\pi)\\[1.4em] &=60\pi. \end{aligned}

Step 4

Check the size and units

The area is $\boxed{60\pi}$ square centimeters, so the answer is C.

Does that size make sense? The fraction $\frac5{12}$ is less than $\frac12$ , so the sector should be less than half the circle. Half of $144\pi$ is $72\pi$ , and $60\pi$ is less, so it fits. Choice D is the whole circle, and choice A, $24\pi$ , is the circumference, a length rather than an area.

Check your understanding:

With the same radius and angle, how long would the arc be?

Choose the fraction before calculating

Every problem here follows the same hybrid routine, part hand work and part Desmos:

Find what the question wants. “How long is the arc?” wants a length, not an angle.
Pick the whole it’s part of. An arc is part of the circumference $2\pi r$ , a sector is part of the area $\pi r^2$ , and an angle is part of $360^\circ$ or $2\pi$ radians.
Write the fraction and simplify it, the way $\frac{150}{360}$ became $\frac5{12}$ .
Finish by hand when the numbers cancel quickly, as $\frac5{12}(144\pi)$ did, and use Desmos when they don’t.
Check that your answer is the thing asked for, in the right units: square units for an area, plain units for a length.

Desmos can’t choose the whole for you. That part is your reasoning. What it does well is the arithmetic on an awkward coefficient, faster and with fewer slips. If the answer should look like $b\pi$ , type in only the expression for $b$ , never a decimal for $\pi$ . Then write the $\pi$ back on, and leave it exact unless the question asks for a decimal.

Try it yourself:

For each practice problem, write down what you’re finding and its whole before you calculate, like “sector out of the circle’s area.”

Practice problems

The first problem runs backward, from a sector’s area to its angle. The fraction still does the work.

Recover an angle from sector area

Practice problem

A circle has radius $10$ centimeters. A sector of the circle has area $35\pi$ square centimeters. What is the measure, in degrees, of the sector’s central angle?

Your answer

Calculator loads as you approach

Compare the sector with the whole circle’s area. This fraction is quickest by hand.

Use a radian fraction with circumference

Practice problem

A circle has circumference $35$ meters. A central angle measuring $\frac{2\pi}{5}$ radians intercepts minor arc $\widehat{MN}$ . What is the length, in meters, of minor arc $\widehat{MN}$ ?

Calculator loads as you approach

Compare the angle with a full turn of

2\pi

radians. This fraction is quickest by hand.

Find a sector from radians

Practice problem

A circle has radius $14$ inches. A sector has a central angle of $\frac{5\pi}{9}$ radians. The area of the sector can be written as $b\pi$ square inches, where $b$ is a constant. What is the value of $b$ ?

Your answer

Calculator loads as you approach

Type only the coefficient, (1/2)*14^2*(5/9), then click the fraction-conversion icon beside the decimal answer.

Finish the lesson

3 practice examples left

Finish the remaining questions correctly to complete this lesson.

Quick recap

A central angle of $\theta^\circ$ takes up $\frac{\theta}{360}$ of the circle.
The arc is that fraction of the circumference, and the sector is that fraction of the area.
An arc’s degree measure is an angle. Its length also depends on how big the circle is.
One full turn is $360^\circ=2\pi$ radians, so $180^\circ=\pi$ radians.
In standard position, an angle starts on the positive $x$ -axis, and positive angles turn counterclockwise.
In radians only, $s=r\theta$ and $A_{\text{sector}}=\frac12r^2\theta$ .
Choose the fraction yourself. When the arithmetic is awkward, let Desmos find the coefficient, and keep $\pi$ exact unless the question asks for a decimal.
Before you answer, check whether the question wants a length, an area, an angle, or a coefficient.

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