Radian Measure

Up to now you’ve measured angles in degrees, where a full turn is $360^\circ$. That number is a choice someone made long ago. Radians measure an angle by the length of arc it cuts off on a circle, which makes many formulas simpler and is the unit used in calculus, physics, and engineering. From here on, trig in this course uses radians unless it says degrees.

Key ideas

What a radian is

Draw a unit circle (radius $1$) with its centre at the vertex of an angle. The radian measure of the angle is the length of the arc that the angle cuts off (the arc it subtends).

So an angle of $1$ radian cuts off an arc exactly as long as the radius. That’s a little less than $60^\circ$: $1 \text{ rad} \approx 57.3^\circ$.

On a unit circle, the angle in radians equals the arc length. On a circle of radius $r$, the arc is $r$ times as long.

A full turn is 2π radians

The whole unit circle has circumference $2\pi(1) = 2\pi$, so a full turn is $2\pi$ radians. Half a turn is $\pi$ radians:

$$360^\circ = 2\pi \text{ rad} \qquad\qquad 180^\circ = \pi \text{ rad}$$

Converting between degrees and radians

Everything comes from $180^\circ = \pi$ rad:

$$\text{radians} = \text{degrees} \times \frac{\pi}{180^\circ} \qquad\qquad \text{degrees} = \text{radians} \times \frac{180^\circ}{\pi}$$

These angles, the special angles and their multiples, come up so often that it’s worth knowing them by heart:

Degrees	$0^\circ$	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$135^\circ$	$150^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
Radians	$0$	$\tfrac{\pi}{6}$	$\tfrac{\pi}{4}$	$\tfrac{\pi}{3}$	$\tfrac{\pi}{2}$	$\tfrac{2\pi}{3}$	$\tfrac{3\pi}{4}$	$\tfrac{5\pi}{6}$	$\pi$	$\tfrac{3\pi}{2}$	$2\pi$

A shortcut: think of $\pi$ as “half a turn”. Then $\tfrac{\pi}{2}$ is a quarter turn ($90^\circ$), $\tfrac{\pi}{3}$ is a third of half a turn ($60^\circ$), and so on.

Exact values and decimals

An angle in radians can be written exactly, in terms of $\pi$, or as a decimal (a rational approximation):

$$\frac{\pi}{3} \approx 1.05 \text{ rad} \qquad\qquad 2\pi \approx 6.28 \text{ rad}$$

When an angle has no unit written, it means radians. So $\theta = 2$ means $2$ radians (about $114.6^\circ$), not $2^\circ$.

Arc length

On a circle of radius $r$, an angle of $\theta$ radians cuts off an arc of length

$$a = r\theta$$

Why: the arc is the fraction $\dfrac{\theta}{2\pi}$ of the whole circumference $2\pi r$, and $\dfrac{\theta}{2\pi} \times 2\pi r = r\theta$. This simple formula only works with $\theta$ in radians, which is one big reason radians are used. The arc $a$ comes out in the same unit as $r$.

Angular velocity

Angular velocity, $\omega$ (the Greek letter omega), is the angle turned per unit of time:

$$\omega = \frac{\theta}{t}$$

It’s usually given in radians per second (rad/s). Spinning things are often described in revolutions per minute (rpm). To convert, use $1$ revolution $= 2\pi$ rad and $1$ min $= 60$ s.

A point at distance $r$ from the centre travels an arc of $r\theta$ in time $t$, so its speed along the circle is

$$v = r\omega$$

Every point on a spinning wheel has the same angular velocity, but points farther from the centre move faster.

Worked examples

Example 1: Degrees to radians

Convert to radians. Give exact answers for (a) to (c), and a decimal to $2$ decimal places for (d).

• (a) $75^\circ$
• (b) $210^\circ$
• (c) $330^\circ$
• (d) $40^\circ$

Solution. Multiply by $\dfrac{\pi}{180^\circ}$ and simplify the fraction.

(a) $75^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{75\pi}{180} = \dfrac{5\pi}{12}$

(b) $210^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{210\pi}{180} = \dfrac{7\pi}{6}$

(c) $330^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{330\pi}{180} = \dfrac{11\pi}{6}$

(d) $40^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{2\pi}{9} \approx 0.70$ rad

Check (b): $210^\circ$ is a bit more than half a turn, and $\tfrac{7\pi}{6}$ is a bit more than $\pi$. ✓

Example 2: Radians to degrees

Convert to degrees: (a) $\dfrac{3\pi}{4}$ (b) $\dfrac{5\pi}{3}$ (c) $2.5$ rad, to $1$ decimal place.

Solution. Multiply by $\dfrac{180^\circ}{\pi}$.

(a) $\dfrac{3\pi}{4} \times \dfrac{180^\circ}{\pi} = \dfrac{3 \times 180^\circ}{4} = 135^\circ$

(b) $\dfrac{5\pi}{3} \times \dfrac{180^\circ}{\pi} = \dfrac{5 \times 180^\circ}{3} = 300^\circ$

(c) $2.5 \times \dfrac{180^\circ}{\pi} \approx 143.2^\circ$

When the angle contains $\pi$, the $\pi$‘s cancel and you get an exact number of degrees. In (c) there’s no $\pi$ to cancel, so the answer is a decimal.

Example 3: Arc length

(a) A pizza with radius $20$ cm is cut into $12$ equal slices. How long is the crust on one slice?

(b) On a circular track of radius $8$ m, a runner covers an arc of $14$ m. Through what angle has she turned, in radians and in degrees?

Solution.

(a) Each slice has central angle $\dfrac{2\pi}{12} = \dfrac{\pi}{6}$. Then

$$a = r\theta = 20 \times \frac{\pi}{6} = \frac{10\pi}{3} \approx 10.47 \text{ cm}$$

(b) Solve $a = r\theta$ for $\theta$:

$$\theta = \frac{a}{r} = \frac{14}{8} = 1.75 \text{ rad} \approx 1.75 \times \frac{180^\circ}{\pi} \approx 100.3^\circ$$

Check (b): $1.75$ rad is a bit more than $\tfrac{\pi}{2} \approx 1.57$, and $100.3^\circ$ is a bit more than $90^\circ$. ✓

Example 4: A wind turbine

The blades of a wind turbine are $45$ m long and turn at $15$ rpm.

(a) Find the angular velocity in radians per second.

(b) How fast is the tip of a blade moving, in m/s and in km/h?

Solution.

(a) Each revolution is $2\pi$ rad, and a minute is $60$ s:

$$\omega = \frac{15 \times 2\pi \text{ rad}}{60 \text{ s}} = \frac{\pi}{2} \text{ rad/s} \approx 1.57 \text{ rad/s}$$

(b) The tip is $45$ m from the centre:

$$v = r\omega = 45 \times \frac{\pi}{2} = 22.5\pi \approx 70.7 \text{ m/s}$$

To change m/s to km/h, multiply by $3.6$: about $254$ km/h. The blades look slow from the ground, but their tips are moving faster than a highway car.

Common mistakes

Calculator in the wrong mode. From now on, most questions are in radians. Check the mode before every calculation: $\sin 1$ should be about $0.841$, not $0.0175$.

Using degrees in the arc length formula. $a = r\theta$ only works when $\theta$ is in radians. With $\theta = 60^\circ$ and $r = 5$, the arc is $5 \times \tfrac{\pi}{3} \approx 5.24$, not $5 \times 60 = 300$.

Thinking π equals 180. $\pi$ is a number, about $3.14$. It’s true that $\pi$ radians is $180^\circ$, the same way $1$ km is $1000$ m, but $\pi$ itself is not $180$.

Losing the π when converting. $210^\circ$ is $\tfrac{7\pi}{6}$, not $\tfrac{7}{6}$. The fraction $\tfrac{7}{6}$ on its own is about $1.17$ rad, a completely different angle.

Forgetting a factor in rpm conversions. Turning rpm into rad/s needs both changes: multiply by $2\pi$ (radians per revolution) and divide by $60$ (seconds per minute).

Leaving fractions unsimplified. $\tfrac{75\pi}{180}$ is correct but hard to use. Simplify to $\tfrac{5\pi}{12}$.

Practice

1. (Warm-up) Convert to radians, exactly: (a) $45^\circ$ (b) $120^\circ$ (c) $270^\circ$

Solution

(a) $45^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{\pi}{4}$

(b) $120^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{2\pi}{3}$

(c) $270^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{3\pi}{2}$

2. (Warm-up) Convert to degrees: (a) $\dfrac{\pi}{5}$ (b) $\dfrac{7\pi}{4}$ (c) $\dfrac{11\pi}{6}$

Solution

(a) $\dfrac{180^\circ}{5} = 36^\circ$

(b) $\dfrac{7 \times 180^\circ}{4} = 315^\circ$

(c) $\dfrac{11 \times 180^\circ}{6} = 330^\circ$

3. (Warm-up) Which angle is larger, $3$ rad or $170^\circ$? Explain.

Solution

$3 \times \dfrac{180^\circ}{\pi} \approx 171.9^\circ$, so $3$ rad is slightly larger than $170^\circ$.

You can also see it without a calculator: $\pi \approx 3.14$ rad is $180^\circ$, and $3$ is just under $\pi$, so $3$ rad is just under $180^\circ$.

4. (Core) (a) Convert $1.2$ rad to degrees, to $1$ decimal place. (b) Convert $252^\circ$ to radians, exactly and as a decimal to $2$ decimal places.

Solution

(a) $1.2 \times \dfrac{180^\circ}{\pi} \approx 68.8^\circ$

(b) $252^\circ \times \dfrac{\pi}{180^\circ} = \dfrac{252\pi}{180} = \dfrac{7\pi}{5} \approx 4.40$ rad

5. (Core) The minute hand of a clock is $12$ cm long. How far does its tip travel in $20$ minutes? Give an exact answer and a decimal.

Solution

In $20$ minutes the hand turns $\tfrac{20}{60} = \tfrac{1}{3}$ of a revolution, which is $\tfrac{1}{3} \times 2\pi = \tfrac{2\pi}{3}$ rad.

$$a = r\theta = 12 \times \frac{2\pi}{3} = 8\pi \approx 25.13 \text{ cm}$$

6. (Core) A sector of a circle has radius $6$ cm and arc length $15$ cm. Find its central angle in radians and in degrees (to $1$ decimal place).

Solution

$$\theta = \frac{a}{r} = \frac{15}{6} = 2.5 \text{ rad}$$

In degrees: $2.5 \times \dfrac{180^\circ}{\pi} \approx 143.2^\circ$.

7. (Core) A bike wheel with a diameter of $70$ cm turns $3$ times every second.

• (a) Find its angular velocity in rad/s.
• (b) How fast is the bike moving, in m/s and in km/h? (The bike moves forward as fast as a point on the tire’s edge moves around the wheel.)

Solution

(a) $\omega = 3 \times 2\pi = 6\pi \approx 18.85$ rad/s.

(b) The radius is $35$ cm $= 0.35$ m, so

$$v = r\omega = 0.35 \times 6\pi = 2.1\pi \approx 6.60 \text{ m/s}$$

That’s $6.60 \times 3.6 \approx 23.8$ km/h.

8. (Challenge) Earth turns once on its axis every $24$ hours, and its radius is about $6371$ km.

• (a) Find Earth’s angular velocity in rad/h.
• (b) How fast is a point on the equator moving because of Earth’s rotation, in km/h?

Solution

(a) $\omega = \dfrac{2\pi}{24} = \dfrac{\pi}{12} \approx 0.262$ rad/h.

(b) $v = r\omega = 6371 \times \dfrac{\pi}{12} \approx 1668$ km/h.

People on the equator are moving at well over the speed of sound, and they don’t feel it, because everything around them is moving too.

9. (Challenge) A merry-go-round turns at $8$ rpm. Maya sits $1$ m from the centre, and her brother sits $2.5$ m from the centre.

• (a) Find their angular velocity in rad/s.
• (b) Find each rider’s speed. Who is moving faster, and why, if they turn through the same angle?

Solution

(a) Both riders turn through the same angle in the same time:

$$\omega = \frac{8 \times 2\pi}{60} = \frac{4\pi}{15} \approx 0.838 \text{ rad/s}$$

(b) Maya: $v = 1 \times \dfrac{4\pi}{15} \approx 0.84$ m/s. Her brother: $v = 2.5 \times \dfrac{4\pi}{15} = \dfrac{2\pi}{3} \approx 2.09$ m/s.

Her brother moves faster. In each turn he travels around a bigger circle (the arc for the same angle is $r\theta$, which grows with $r$), but in the same amount of time.