Math Core

How far does a point on a bicycle tire travel when the wheel turns a third of the way around? How fast is the tip of a fan blade moving? Radians make questions like these almost effortless, because a radian measure is already a ratio of lengths. In this lesson you'll turn angles into distances and rotation rates into speeds.

Arc length

Recall the definition of radian measure: a central angle $\theta$ that cuts off an arc of length $s$ on a circle of radius $r$ has measure $\theta = \dfrac{s}{r}$ . Multiply both sides by $r$ and you get a formula for the arc.

A central angle θ (in radians) on a circle of radius r intercepts an arc of length s = rθ.

Arc length

On a circle of radius $r$ , a central angle of $\theta$ radians intercepts an arc of length

s = r\theta.

The arc length $s$ has the same units as $r$ .

This is just proportional reasoning. The whole circle is an angle of $2\pi$ with arc length $2\pi r$ . An angle of $\theta$ is the fraction $\dfrac{\theta}{2\pi}$ of the circle, so its arc is $\dfrac{\theta}{2\pi} \cdot 2\pi r = r\theta$ .

Worked example: Finding an arc length

A circle has radius $8$ cm. Find the length of the arc cut off by a central angle of $\dfrac{3\pi}{4}$ .

s = r\theta = 8 \cdot \frac{3\pi}{4} = 6\pi \text{ cm} \approx 18.85 \text{ cm}

Common mistake

The formula $s = r\theta$ only works when $\theta$ is in radians. If you plug in degrees, you get nonsense: $8 \cdot 135 = 1080$ cm is far longer than the whole circle, whose circumference is only about $50$ cm. Convert degrees to radians first.

Worked example: An angle given in degrees

A pendulum $30$ inches long swings through an angle of $24^\circ$ . How far does the tip travel along its arc? Round to the nearest tenth.

Convert first: $24^\circ \cdot \dfrac{\pi}{180^\circ} = \dfrac{2\pi}{15}$ radians. Then

s = r\theta = 30 \cdot \frac{2\pi}{15} = 4\pi \approx 12.6 \text{ inches}.

You can also run the formula backward. If you know the arc and the radius, then $\theta = \dfrac{s}{r}$ gives the angle in radians. For example, an arc of $15$ m on a circle of radius $6$ m corresponds to a central angle of $\dfrac{15}{6} = 2.5$ radians.

Area of a sector

A sector is the pie-slice region bounded by two radii and their arc. The same proportional reasoning works for area: the sector is $\dfrac{\theta}{2\pi}$ of the circle's area $\pi r^2$ , so

A = \frac{\theta}{2\pi} \cdot \pi r^2 = \frac{1}{2} r^2 \theta \qquad (\theta \text{ in radians}).

For instance, a sector with radius $6$ in and angle $\dfrac{\pi}{3}$ has area $\dfrac{1}{2} \cdot 36 \cdot \dfrac{\pi}{3} = 6\pi$ square inches.

Angular speed and linear speed

When something spins, there are two natural ways to describe how fast it moves.

Definition

Angular and linear speed

Suppose a point moves along a circle of radius $r$ and sweeps out an angle $\theta$ (in radians) while traveling an arc of length $s$ in time $t$ .

Its angular speed is $\omega = \dfrac{\theta}{t}$ , the angle turned per unit of time.
Its linear speed is $v = \dfrac{s}{t}$ , the distance traveled per unit of time.

The symbol $\omega$ is the Greek letter omega. Angular speed is measured in units like radians per second or revolutions per minute (rpm). Linear speed is measured in units like feet per second or miles per hour.

Every point on a spinning wheel has the same angular speed, since the whole wheel turns together. But points farther from the center travel bigger circles in the same time, so they move faster. Dividing $s = r\theta$ by $t$ shows exactly how the two speeds are related.

Linking the two speeds

v = r\omega \qquad (\omega \text{ in radians per unit time})

Rotation rates are often given in revolutions. Since one revolution is $2\pi$ radians, multiply revolutions by $2\pi$ to get radians.

Worked example: A spinning wheel

A bicycle wheel has a diameter of $26$ inches and turns at $200$ revolutions per minute.

Find its angular speed in radians per minute.
Find the linear speed of a point on the tire, in inches per minute and in miles per hour.

Solutions.

$\omega = 200 \dfrac{\text{rev}}{\text{min}} \cdot \dfrac{2\pi \text{ rad}}{1 \text{ rev}} = 400\pi$ radians per minute.
The radius is $13$ inches, so $v = r\omega = 13 \cdot 400\pi = 5200\pi \approx 16{,}336$ inches per minute. To get miles per hour, multiply by $60$ minutes per hour and divide by $63{,}360$ inches per mile:

\frac{5200\pi \cdot 60}{63{,}360} \approx 15.5 \text{ miles per hour}.

A point on the tire moves at the bike's speed, so the bike is going about $15.5$ mph.

Worked example: Comparing two riders

On a merry-go-round turning at $\dfrac{\pi}{4}$ radians per second, Ana sits $2$ m from the center and Ben sits $5$ m from the center. How much faster is Ben moving?

Both have the same angular speed, $\omega = \dfrac{\pi}{4}$ rad/s.

Ana: $v = 2 \cdot \dfrac{\pi}{4} = \dfrac{\pi}{2} \approx 1.57$ m/s.
Ben: $v = 5 \cdot \dfrac{\pi}{4} = \dfrac{5\pi}{4} \approx 3.93$ m/s.

Ben moves $\dfrac{5\pi}{4} - \dfrac{\pi}{2} = \dfrac{3\pi}{4} \approx 2.36$ m/s faster, even though both complete a lap in the same $8$ seconds.

Tip

Track units like fractions and cancel them. Revolutions cancel with "rev" in $\dfrac{2\pi \text{ rad}}{1 \text{ rev}}$ , and minutes cancel with $\dfrac{1 \text{ min}}{60 \text{ s}}$ . Radians have no units, so in $v = r\omega$ the answer takes the units of $r$ divided by time.

Practice

Practice 1

A circle has radius $5$ cm. Find the length of the arc cut off by a central angle of $2$ radians, in centimeters.