Radians, Arc Length and Circular Motion

What a radian measures

For a circle of radius $r>0$, an arc of length $s$ subtends a central angle whose radian measure is $\theta=s/r$. An arc as long as the radius therefore subtends one radian. Doubling both the arc length and the radius leaves the angle unchanged.

Because circumference is $2\pi r$, one complete turn has radian measure $2\pi r/r=2\pi$. The number $\pi$ here is not a label for degrees. It is part of an exact real-number measure of the angle.

One radian: arc length equals radius

Diagram explained

The source marks an arc length equal to the radius and labels the central angle one radian. In the general formula theta equals s divided by r, this is the special case s equals r. The arc is curved, whereas the radius runs straight from the centre to the circle.

Radians are a ratio of two lengths expressed in the same unit. The ratio is dimensionless, but writing “rad” is useful for distinguishing an angle from another dimensionless quantity. A bare number inside a trigonometric function is conventionally interpreted in radians unless another unit is stated.

Converting degrees and radians

A full turn is both $360^\circ$ and $2\pi$ radians. Use that equality to construct the conversion factor, rather than guessing where $\pi$ belongs:

Worked conversions

For $150^\circ$, the radian measure is $150\pi/180=5\pi/6$. Conversely, $7\pi/6$ radians is $210^\circ$. A negative angle follows the same conversion: $-45^\circ=-\pi/4$.

Keep $\pi$ in an exact expression until a decimal answer is required. Replacing it with 3.14 early can compound rounding in subsequent calculations.

Directed angles and complete turns

In standard coordinate orientation, counterclockwise rotation is positive and clockwise rotation is negative. Angles differing by an integer multiple of $2\pi$ have the same terminal direction. They are coterminal.

For example, $13\pi/6$ and $\pi/6$ finish in the same direction because their difference is one full turn. They do not describe the same amount of rotation. A wheel that turns $13\pi/6$ travels farther around its rim than one that turns $\pi/6$.

Use the actual angle travelled in a motion problem, not merely a coterminal representative on the unit circle. If the object reverses direction, total distance travelled depends on every part of the motion; net angular displacement alone is not sufficient.

Finding arc length

Rearranging the radian definition gives the length of a circular arc:

For a directed angular displacement, $r\theta$ is a signed arc displacement; the corresponding single-direction distance is $r|\theta|$. Here we use non-negative angles when calculating a physical length.

Worked example: a 150° arc

A circle has radius 12 cm. First convert $150^\circ$ to $5\pi/6$ radians. Then

The exact answer is $10\pi$ cm, about 31.42 cm. A check based on proportions gives the same result: $150/360$ of the circumference $24\pi$ is $10\pi$. Substituting 150 directly into $s=r\theta$ would treat 150 as a radian measure and give the wrong length.

Sector area and chord length are different quantities

A sector is the region enclosed by two radii and their connecting arc. Its fraction of the circle is $\theta/(2\pi)$, so multiplying that fraction by $\pi r^2$ gives

For the 12 cm radius and $150^\circ$ angle above, the sector area is $\tfrac12(12)^2(5\pi/6)=60\pi$ cm². Area uses square units; arc length uses length units.

The straight connection between the endpoints

A chord is the straight segment joining two points on a circle. Splitting the isosceles triangle formed by the radii gives

For $0\le\theta\le2\pi$, this is the non-negative chord length between the two endpoints. In the example, $c=24\sin75^\circ=6(\sqrt6+\sqrt2)$ cm, about 23.18 cm. It is shorter than the $10\pi$ cm arc. An arc, a chord and a sector can share endpoints or boundaries while measuring entirely different things.

For a simple sector, use an angle no larger than one complete turn. Applying the sector formula to several revolutions counts swept area repeatedly; it does not describe a larger region inside the same circle.

Angular speed and speed around a circle

Angular speed measures the angle turned per unit time. For uniform rotation, $\omega=\Delta\theta/\Delta t$. A point at distance $r$ from the axis travels along a circle, so its tangential speed is

Here $\omega$ is a non-negative angular speed in radians per unit time. For signed angular velocity, the corresponding speed is $r|\omega|$. Points on the same rigid rotating object share an angular speed, but points farther from the axis have a larger tangential speed.

Worked example: converting revolutions per minute

A wheel of radius 0.25 m rotates uniformly at 120 revolutions per minute. Convert revolutions to radians and minutes to seconds:

The rim speed is about 3.14 m/s. For a wheel rolling without slipping, this is also the speed of its centre relative to the ground. That additional interpretation needs the no-slip condition; a wheel spinning in place has the same rim speed relative to its centre without travelling forward.

Period, frequency and circular coordinates

The period $T$ is the time for one revolution. The frequency $f$ is the number of revolutions per unit time. For uniform rotation,

A point travelling counterclockwise around a circle centred at the origin can be described by $x(t)=r\cos(\omega t+\phi)$ and $y(t)=r\sin(\omega t+\phi)$. The phase $\phi$ identifies the starting direction. Each coordinate varies sinusoidally even though the point’s distance from the centre stays constant.

Check the quantity before calculating

Use $s=r\theta$ for a distance along the circle, $\tfrac12r^2\theta$ for sector area, and $2r\sin(\theta/2)$ for the straight chord. In each formula, name the angle and identify whether the answer should have length or area units.

For motion, distinguish angular displacement from the total turn travelled and angular speed from tangential speed. Match each formula to the quantity it measures before substituting values.

Related topics

Apply circular motion to Sinusoidal Modelling. To locate points by their radius and angle directly, continue to Polar Coordinates and Trigonometry.

Practice questions

1. A circle has radius 6 cm and central angle $2/3$ radians. Find the arc length and sector area.
2. A wheel rotates at 90 revolutions per minute. Find its angular speed in radians per second.
3. Why can an angle of $13\pi/6$ not simply be replaced by $\pi/6$ when finding distance travelled around a circle?

Sources and further reading

Further explanations of the definitions and conventions used here:

• OpenStax, Precalculus 2e: Radians, arc length and angular speed.
• Wolfram Language: Trigonometric Functions. Used for checking exact values, expressions and solutions.