Math Core

An arc's measure tells you what fraction of a full turn it covers, in degrees. Its length tells you how far you'd walk along it, in inches or meters. This lesson connects the two, and then uses that connection to define a new way to measure angles: radians.

Arc length as a fraction of the circumference

An arc of $90^\circ$ is $\dfrac{90}{360} = \dfrac{1}{4}$ of the full circle, so its length is $\dfrac{1}{4}$ of the circumference. The same reasoning works for any central angle.

Arc length (degrees)

In a circle of radius $r$ , the length $s$ of an arc with central angle $\theta^\circ$ is

$s = \dfrac{\theta}{360} \cdot 2\pi r.$

A 120° arc in a circle of radius 4 is one third of the circumference.

Worked example: Finding an arc length

Find the length of a $120^\circ$ arc in a circle of radius $4$ cm.

$s = \dfrac{120}{360} \cdot 2\pi(4) = \dfrac{1}{3} \cdot 8\pi = \dfrac{8\pi}{3} \text{ cm} \approx 8.38 \text{ cm}.$

Worked example: Working backward

An arc in a circle of radius $10$ has length $5\pi$ . Find the measure of its central angle.

Set up the formula and solve for $\theta$ :

$5\pi = \dfrac{\theta}{360} \cdot 20\pi \quad\Longrightarrow\quad \dfrac{\theta}{360} = \dfrac{5\pi}{20\pi} = \dfrac{1}{4} \quad\Longrightarrow\quad \theta = 90^\circ.$

Common mistake

Arc measure and arc length are different. Two $60^\circ$ arcs have the same measure, but in circles of radius $1$ and radius $100$ their lengths are $\frac{\pi}{3}$ and $\frac{100\pi}{3}$ . Measure uses degrees; length uses units of distance.

Radians

Here is a remarkable fact. All circles are similar, so if you divide an arc's length by the radius, the result depends only on the angle, not on the size of the circle. That ratio is a natural way to measure the angle.

Definition

Radian

The radian measure of a central angle is the length of its intercepted arc divided by the radius:

$\theta = \dfrac{s}{r}.$

An angle of $1$ radian intercepts an arc exactly as long as the radius.

A full circle has arc length $2\pi r$ , so a full turn measures $\dfrac{2\pi r}{r} = 2\pi$ radians. Since a full turn is also $360^\circ$ ,

$360^\circ = 2\pi \text{ radians} \qquad\text{and}\qquad 180^\circ = \pi \text{ radians}.$

That second equation is the only conversion fact you need.

Converting between degrees and radians

$\text{radians} = \text{degrees} \cdot \dfrac{\pi}{180} \qquad\qquad \text{degrees} = \text{radians} \cdot \dfrac{180}{\pi}$

degrees	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
radians	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$

One radian is $\dfrac{180^\circ}{\pi} \approx 57.3^\circ$ , a little less than $60^\circ$ .

Why bother with a second unit? Degrees are an arbitrary choice: someone long ago decided a full turn should be $360$ parts. Radians come straight from the circle itself, which makes many formulas simpler. You'll see that in a moment with arc length, again with sector area, and constantly in trigonometry and calculus. Notice also that radian answers are often written without a unit: "an angle of $2$ " means $2$ radians.

Worked example: Converting

Convert $150^\circ$ to radians.
Convert $\dfrac{5\pi}{4}$ radians to degrees.

Solutions.

$150 \cdot \dfrac{\pi}{180} = \dfrac{150\pi}{180} = \dfrac{5\pi}{6}$ radians.
$\dfrac{5\pi}{4} \cdot \dfrac{180}{\pi} = \dfrac{5 \cdot 180}{4} = 225^\circ$ .

Arc length with radians

Rearranging $\theta = \dfrac{s}{r}$ gives the simplest arc length formula of all.

Arc length (radians)

If the central angle $\theta$ is in radians, then $s = r\theta$ .

Worked example: Using s = rθ

A pendulum $2$ meters long swings through an angle of $0.6$ radians. How far does the tip travel along its arc?

$s = r\theta = 2(0.6) = 1.2 \text{ meters}.$

Tip

Only use $s = r\theta$ when $\theta$ is in radians. If the angle is in degrees, either use the $\dfrac{\theta}{360}$ formula or convert to radians first. The two methods always agree: for $120^\circ$ and $r = 4$ , $\dfrac{2\pi}{3} \cdot 4 = \dfrac{8\pi}{3}$ .

Practice

Practice 1

Find the length of a $40^\circ$ arc in a circle of radius $9$ . Give your answer in terms of $\pi$ (type pi).