Degrees are a human choice. The number 360 comes from ancient astronomy and divides nicely, but it has nothing to do with the circle itself. The radian does. It measures an angle by the arc it cuts off, using the circle's own radius as the ruler.
That one idea explains why $latex \pi$ shows up in every trigonometry formula, why arc length and sector area formulas are so short, and why calculus insists on radians. Drag the point below and watch the arc being laid out along the ruler.
What is a radian?
One radian is the angle at the center of a circle that cuts off an arc exactly as long as the radius. Take the radius, bend it along the circle, and the angle it covers is 1 radian. In the tool above, every section of the ruler is one radius long, so the number of sections the arc covers is the angle in radians.
For any central angle $latex \theta$ that cuts off an arc of length $latex s$ on a circle of radius $latex r$:
$$\theta=\frac{s}{r}$$
This is a ratio of two lengths, so a radian has no units. Writing "rad" is optional and only a reminder.
Why a full turn is 2π radians
The circumference is $latex 2\pi r$, which is $latex 2\pi$ radius lengths. So a full turn is $latex 2\pi$ radians, a half turn is $latex \pi$ and a quarter turn is $latex \frac{\pi}{2}$. That is why the ruler in the tool ends at $latex \pi$ on its first row and $latex 2\pi$ on its second.
Since $latex \pi\text{ rad}=180^{\circ}$, one radian is a little less than 60 degrees:
$$1\text{ rad}=\frac{180^{\circ}}{\pi}\approx 57.2958^{\circ}$$
Converting between degrees and radians
Multiply by the right fraction so that the unwanted unit cancels:
$$\text{radians}=\text{degrees}\times\frac{\pi}{180}\qquad\qquad \text{degrees}=\text{radians}\times\frac{180}{\pi}$$
For long drills on conversions, see our articles on degrees to radians and radians to degrees. Here the goal is to understand what the numbers mean.
Special angles on the unit circle
The unit circle is the circle of radius 1 centered at the origin. A handful of angles come up constantly, and their sine and cosine are exact values worth knowing.
| Degrees | Radians | cos | sin |
|---|---|---|---|
| 0° | $latex 0$ | $latex 1$ | $latex 0$ |
| 30° | $latex \frac{\pi}{6}$ | $latex \frac{\sqrt{3}}{2}$ | $latex \frac{1}{2}$ |
| 45° | $latex \frac{\pi}{4}$ | $latex \frac{\sqrt{2}}{2}$ | $latex \frac{\sqrt{2}}{2}$ |
| 60° | $latex \frac{\pi}{3}$ | $latex \frac{1}{2}$ | $latex \frac{\sqrt{3}}{2}$ |
| 90° | $latex \frac{\pi}{2}$ | $latex 0$ | $latex 1$ |
| 120° | $latex \frac{2\pi}{3}$ | $latex -\frac{1}{2}$ | $latex \frac{\sqrt{3}}{2}$ |
| 135° | $latex \frac{3\pi}{4}$ | $latex -\frac{\sqrt{2}}{2}$ | $latex \frac{\sqrt{2}}{2}$ |
| 150° | $latex \frac{5\pi}{6}$ | $latex -\frac{\sqrt{3}}{2}$ | $latex \frac{1}{2}$ |
| 180° | $latex \pi$ | $latex -1$ | $latex 0$ |
| 270° | $latex \frac{3\pi}{2}$ | $latex 0$ | $latex -1$ |
| 360° | $latex 2\pi$ | $latex 1$ | $latex 0$ |
You do not need to memorize the table. For 0°, 30°, 45°, 60° and 90°, the sine values are $latex \frac{\sqrt{0}}{2},\frac{\sqrt{1}}{2},\frac{\sqrt{2}}{2},\frac{\sqrt{3}}{2},\frac{\sqrt{4}}{2}$. The cosines are the same list backwards. The other rows are mirror images of these, with signs set by the quadrant.
Sine and cosine as coordinates
Here is the picture behind the definitions. Start at the point $latex (1,0)$ and move counterclockwise along the unit circle until the arc has length $latex \theta$. You are at the point:
$$P(\theta)=(\cos\theta,\ \sin\theta)$$
The horizontal blue segment in the tool is $latex \cos\theta$ and the vertical red one is $latex \sin\theta$. Because the point lies on a circle of radius 1, $latex x^2+y^2=1$, which gives the identity $latex \cos^2\theta+\sin^2\theta=1$.
- Quadrant I (0 to $latex \frac{\pi}{2}$): cosine positive, sine positive.
- Quadrant II ($latex \frac{\pi}{2}$ to $latex \pi$): cosine negative, sine positive.
- Quadrant III ($latex \pi$ to $latex \frac{3\pi}{2}$): both negative.
- Quadrant IV ($latex \frac{3\pi}{2}$ to $latex 2\pi$): cosine positive, sine negative.
On a circle of radius $latex r$, the same point is $latex (r\cos\theta,\ r\sin\theta)$.
Why radians make the formulas simple
With the angle in radians, the arc length and the sector area need no conversion factor:
$$s=r\theta\qquad\qquad A=\frac{1}{2}r^2\theta$$
In degrees the same formulas carry an extra $latex \frac{\pi}{180}$, for example $latex s=\frac{\pi r\theta}{180}$. Calculus goes further: the derivative of $latex \sin x$ is $latex \cos x$ only when $latex x$ is in radians. In degrees an unwelcome factor of $latex \frac{\pi}{180}$ appears every time. Our articles on the arc of a circumference and the area of a circular sector work these formulas in detail.
Radians: worked examples
Solve each problem before opening the solution.
EXAMPLE 1
Convert $latex 150^{\circ}$ to radians and $latex \frac{7\pi}{4}$ to degrees.
Solution
$$150^{\circ}\times\frac{\pi}{180}=\frac{150\pi}{180}=\frac{5\pi}{6}$$
$$\frac{7\pi}{4}\times\frac{180^{\circ}}{\pi}=\frac{7\cdot 180^{\circ}}{4}=315^{\circ}$$
Reduce the fraction of $latex \pi$ as far as possible: $latex \frac{150\pi}{180}$ is $latex \frac{5\pi}{6}$.
EXAMPLE 2
Find the length of the arc on a circle of radius 8 cm for a central angle of $latex \frac{3\pi}{4}$ radians.
Solution
$$s=r\theta=8\cdot\frac{3\pi}{4}=6\pi\approx 18.85$$
The arc is 6π cm long, about 18.85 cm.
EXAMPLE 3
Find the area of a sector of radius 6 m and central angle $latex \frac{2\pi}{3}$.
Solution
$$A=\frac{1}{2}r^2\theta=\frac{1}{2}\cdot 36\cdot\frac{2\pi}{3}=12\pi\approx 37.7$$
The sector has an area of 12π m², about 37.7 m².
EXAMPLE 4
Find the coordinates of the point on a circle of radius 10 centered at the origin at angle $latex \frac{5\pi}{6}$.
Solution
The angle is $latex 150^{\circ}$, in quadrant II. Using the table, $latex \cos\frac{5\pi}{6}=-\frac{\sqrt{3}}{2}$ and $latex \sin\frac{5\pi}{6}=\frac{1}{2}$.
$$(x,y)=\left(10\cdot\left(-\frac{\sqrt{3}}{2}\right),\ 10\cdot\frac{1}{2}\right)=(-8.66,\ 5)$$
The x-coordinate is negative, as expected in the second quadrant.
EXAMPLE 5
The minute hand of a clock is 12 cm long. How far does its tip travel in 20 minutes?
Solution
In 20 minutes the hand turns one third of a full turn: $latex \theta=\frac{1}{3}\cdot 2\pi=\frac{2\pi}{3}$ radians.
$$s=r\theta=12\cdot\frac{2\pi}{3}=8\pi\approx 25.13$$
The tip travels about 25.13 cm.
EXAMPLE 6
A bicycle wheel of radius 0.35 m turns through 5 radians. How far does the bicycle roll, and how many degrees is that?
Solution
$$s=r\theta=0.35\cdot 5=1.75\text{ m}$$
$$5\text{ rad}\times\frac{180^{\circ}}{\pi}\approx 286.48^{\circ}$$
The bicycle rolls 1.75 m, and 5 radians is about 286.48 degrees, a little less than one full turn.
A feel for the size of a radian
It helps to have a few reference sizes in your head, so a wrong answer looks wrong at a glance:
- 0.5 rad is about 28.6° and 1 rad is about 57.3°.
- 2 rad is about 114.6° and 3 rad is about 171.9°, just under a half turn.
- 3.14 rad ($latex \pi$) is exactly a half turn, and 6.28 rad ($latex 2\pi$) is a full turn.
Here is a quick use of that feel. If your calculator is in radian mode and you type sin(30), you get -0.988 instead of 0.5, because 30 radians is more than four full turns. A result that far from 0.5 is the sign that the mode is wrong.
Where you will meet radians next
Once you think in radius lengths, two later topics get easier. In physics, angular speed is measured in radians per second, and a point at distance $latex r$ from the axis moves at speed $latex v=r\omega$. A record turning at $latex 33\frac{1}{3}$ revolutions per minute has $latex \omega=33\frac{1}{3}\cdot\frac{2\pi}{60}\approx 3.49$ rad/s, so a groove 0.1 m from the center moves at about 0.35 m/s.
In calculus, the limit $latex \lim_{x\to 0}\frac{\sin x}{x}=1$ holds only when $latex x$ is in radians. That single fact is what makes the derivative of sine come out as cosine without any extra constants.
Common mistakes
- Calculator in the wrong mode. If your angle is in radians, switch to RAD before pressing sin or cos.
- Writing $latex \pi=180$. The correct statement is $latex \pi\text{ radians}=180^{\circ}$. The number $latex \pi$ is about 3.14.
- Using degrees in $latex s=r\theta$. The formula needs $latex \theta$ in radians, or you must convert first.
- Thinking 1 radian is a tiny angle. It is about 57.3 degrees.
- Losing the sign of the coordinates. Decide the quadrant first, then use the reference angle.
- Leaving fractions unreduced, such as $latex \frac{150\pi}{180}$ instead of $latex \frac{5\pi}{6}$.
Radians: practice problems
Use the formulas and the table above. Open each answer to check.
1. Convert $latex 240^{\circ}$ to radians.
Answer
$latex 240\cdot\frac{\pi}{180}=\frac{4\pi}{3}$ radians.
2. Convert $latex \frac{11\pi}{6}$ radians to degrees.
Answer
$latex \frac{11\cdot 180^{\circ}}{6}=330^{\circ}$.
3. An arc on a circle of radius 5 cm subtends an angle of 1.4 radians. How long is the arc?
Answer
$latex s=5\cdot 1.4=7$ cm.
4. Find the area of a sector with radius 10 cm and central angle $latex \frac{\pi}{5}$.
Answer
$latex A=\frac{1}{2}\cdot 100\cdot\frac{\pi}{5}=10\pi\approx 31.42$ cm².
5. Give the exact coordinates of the point at angle $latex \frac{5\pi}{4}$ on the unit circle.
Answer
$latex \left(-\frac{\sqrt{2}}{2},\ -\frac{\sqrt{2}}{2}\right)$, in quadrant III where both coordinates are negative.
6. A point on the unit circle in quadrant II has x-coordinate $latex -0.6$. Find its y-coordinate and its angle in radians.
Answer
$latex y=\sqrt{1-0.36}=0.8$. The angle is $latex \theta=\arccos(-0.6)\approx 2.2143$ radians, which is about 126.87°.
Frequently asked questions
Why does a full circle have 2π radians?
Because the circumference is $latex 2\pi r$, and each radius length of arc is one radian. A full turn holds $latex 2\pi$ radius lengths.
How many degrees is one radian?
About 57.2958 degrees, from $latex \frac{180}{\pi}$.
Does a radian have a unit?
It is a ratio of two lengths, so it is dimensionless. Writing "rad" tells the reader which angle measure you mean, nothing more.
When should I use degrees and when radians?
Degrees are fine for everyday angles and construction. Use radians for arc length, sector area, trigonometric functions as graphs and anything in calculus or physics.
What does π radians mean on a calculator?
Most calculators have a RAD mode. Typing sin(π/6) in RAD mode returns 0.5, the same as sin(30) in DEG mode.
Related calculators
Practice and check with these tools:
See also
More on angles, circles and the unit circle: