1. Two ways to measure rotation
An angle is, fundamentally, an amount of rotation. You turn a steering wheel a little or a lot; you tilt a picture frame; you point a telescope from one star to another. To describe how much, you pick a unit and count.
Two units dominate. The first is the degree, where a full turn is split into $360$ equal parts. The choice of $360$ is historical — Babylonian astronomers worked in a base-60 system, and $360$ is conveniently close to the number of days in a year while being divisible by almost everything (2, 3, 4, 5, 6, 8, 9, 10, 12, ...). It's arbitrary, but it's been the working unit of navigators, surveyors, and schoolchildren for two thousand years.
The second is the radian. Where degrees ask "how many of the 360 equal slices did we sweep?", radians ask a different and stranger question: "how long is the arc we traced, measured in radii?" That sounds inconvenient — and at first, it is. But it turns out to be the unit nature itself is using, and once you start doing calculus there's no going back.
Use degrees when you're communicating with humans, building things, or reading a compass. Use radians when you're doing analysis, calculus, physics, or anything where the answer is going to be plugged into a formula with $\sin$, $\cos$, or $e^{i\theta}$ in it. Neither unit is "better" — they just live in different rooms.
2. What a radian is
The angle subtended at the centre of a circle by an arc whose length equals the circle's radius. Equivalently: the angle for which $\text{arc length} = \text{radius}$.
This definition is worth pausing on, because it's stranger than it looks. The radian is not a count of "slices of the pie" — it's the ratio of an arc to a radius. The unit comes out of the circle's own geometry; nobody chose it the way 360 was chosen. If aliens reinvented mathematics from scratch, they wouldn't reinvent the degree, but they'd probably end up at the radian.
Picture a circle of radius $r$. Walk along the circumference until you've travelled a distance of exactly $r$. The angle you've swept out, as seen from the centre, is $1$ radian. Walk another $r$, and you've swept $2$ radians. Keep going all the way around — the full circumference is $2\pi r$, so a full turn is $2\pi$ radians.
Because the radian is a ratio of two lengths (arc divided by radius), it's dimensionless. That's why you can drop the unit in formulas — when you write $\sin(\theta)$ and $\theta$ is in radians, $\theta$ is really just a number. This is also why radians sometimes look like they're "missing" a unit; they're not — that's the feature.
3. The conversion
Half a turn — straight across the circle, from the top to the bottom — covers an arc of length $\pi r$ (half the circumference). In radian terms, that's $\pi$ radians. In degree terms, it's $180°$. So:
$$ \pi \text{ rad} = 180° $$Everything else follows. Divide both sides by $180$ to convert one degree:
$$ 1° = \frac{\pi}{180} \text{ rad} \approx 0.01745 \text{ rad} $$Divide both sides by $\pi$ to convert one radian:
$$ 1 \text{ rad} = \frac{180}{\pi}° \approx 57.296° $$The conversion is multiplicative — multiply degrees by $\pi/180$ to get radians, multiply radians by $180/\pi$ to get degrees. Memorize the equivalence $\pi = 180°$ and you'll never lose your way.
Common angles
These come up so often that the radian values are worth knowing on sight. They're the angles you'll see again in the unit circle, in special triangles, and in every trig table from here on.
| Degrees | Radians (exact) | Radians (decimal) | Fraction of a turn |
|---|---|---|---|
| $0°$ | $0$ | $0$ | $0$ |
| $30°$ | $\pi/6$ | $\approx 0.524$ | $1/12$ |
| $45°$ | $\pi/4$ | $\approx 0.785$ | $1/8$ |
| $60°$ | $\pi/3$ | $\approx 1.047$ | $1/6$ |
| $90°$ | $\pi/2$ | $\approx 1.571$ | $1/4$ |
| $180°$ | $\pi$ | $\approx 3.142$ | $1/2$ |
| $270°$ | $3\pi/2$ | $\approx 4.712$ | $3/4$ |
| $360°$ | $2\pi$ | $\approx 6.283$ | $1$ |
When mathematicians write something like $\sin(\pi/3)$, they always mean the argument is in radians. There's no separate "radian mode" in the notation — radians are the default, and degrees are the special case that gets flagged with a little circle.
4. Standard position and coterminal angles
To compare angles cleanly, mathematicians anchor them to a fixed frame. An angle is in standard position when its vertex sits at the origin of the $xy$-plane and its initial side lies along the positive $x$-axis. The terminal side is wherever the rotation lands.
A positive angle rotates the terminal side counter-clockwise; a negative angle rotates it clockwise. So $-\tfrac{\pi}{2}$ rad points straight down (terminal side along the negative $y$-axis), while $+\tfrac{\pi}{2}$ rad points straight up.
Because a full turn brings you back to where you started, infinitely many different angles share the same terminal side. They are called coterminal. Any two coterminal angles differ by an integer number of full turns:
$$ \theta_{\text{coterminal}} = \theta + 2\pi k \quad \text{(radians)}, \qquad \theta_{\text{coterminal}} = \theta + 360°\, k \quad \text{(degrees)} $$for any integer $k$ (positive or negative). For example, $\tfrac{\pi}{3}$, $\tfrac{7\pi}{3}$, and $-\tfrac{5\pi}{3}$ are all coterminal — they all point the same direction.
To find the coterminal angle in $[0, 2\pi)$, add or subtract $2\pi$ until you land in that interval. In degrees, add or subtract $360°$ until you're in $[0°, 360°)$. This is how trig calculators avoid having to compute $\sin(1{,}000{,}000)$ from scratch.
5. Why calculus prefers radians
Up to here, the choice between degrees and radians might look like taste — a matter of which numbers you'd rather write down. But the moment derivatives enter, the choice becomes forced. Working in radians, the two foundational derivatives of trigonometry are spectacularly clean:
$$ \frac{d}{d\theta}\sin\theta = \cos\theta, \qquad \frac{d}{d\theta}\cos\theta = -\sin\theta $$That's it. The derivative of $\sin$ is $\cos$. No constants, no conversion factors, no apologies. This is what makes power series for $\sin$ and $\cos$ work out so neatly, makes Euler's formula $e^{i\theta} = \cos\theta + i\sin\theta$ hold without correction, and makes simple harmonic motion in physics look like $\ddot x = -\omega^2 x$ instead of something cluttered.
If you insist on degrees, the derivative grows a wart:
$$ \frac{d}{d\theta}\sin\theta^\circ = \frac{\pi}{180}\cos\theta^\circ $$That $\pi/180$ now appears every time you differentiate a trig function — and once, twice, three times for higher derivatives. The whole apparatus of calculus becomes harder to read for no reason except that you picked the wrong unit.
The underlying fact is the limit
$$ \lim_{\theta \to 0} \frac{\sin\theta}{\theta} = 1 $$which is true only when $\theta$ is in radians. In degrees the limit equals $\pi/180$, and that constant propagates through every derivative that follows. Radians are the unit in which sine is, to first order, the identity function near zero — and that is what makes them "natural" in a technical, not a poetic, sense.
Don't think of radians as "degrees with $\pi$ stuck on". Think of them as the unit in which $\sin(\theta) \approx \theta$ for small $\theta$ — and therefore the unit in which the calculus of trig functions stops needing apologies.
6. Arc length and sector area
The radian's geometric definition pays off again here. If a circle has radius $r$ and you sweep out a central angle of $\theta$ radians, the arc length is simply
$$ s = r\theta $$This isn't a coincidence — it's the definition unwound. One radian gave you an arc of length $r$; $\theta$ radians give you $\theta$ copies of that arc. The "formula" is barely a formula.
The area of the corresponding pie-slice (a circular sector) is
$$ A = \tfrac{1}{2} r^2 \theta $$Again with $\theta$ in radians. The factor $\tfrac{1}{2}$ comes from integration — the same $\tfrac{1}{2}$ that shows up in the area of a triangle.
Compare with degrees
If you stubbornly use degrees, you have to convert on the fly. The arc length becomes
$$ s = r \cdot \theta_\text{deg} \cdot \frac{\pi}{180} = \frac{\pi r \theta_\text{deg}}{180} $$and the sector area becomes
$$ A = \frac{1}{2} r^2 \cdot \theta_\text{deg} \cdot \frac{\pi}{180} = \frac{\pi r^2 \theta_\text{deg}}{360} $$The $\pi$ and the $180$ aren't doing any mathematical work — they're paying rent for keeping the wrong unit around. Switch to radians, and the rent disappears.
| Quantity | Radians | Degrees |
|---|---|---|
| Arc length | $s = r\theta$ | $s = \dfrac{\pi r \theta_\text{deg}}{180}$ |
| Sector area | $A = \tfrac{1}{2}r^2\theta$ | $A = \dfrac{\pi r^2 \theta_\text{deg}}{360}$ |
| Full-circle check ($\theta = 2\pi$ or $360°$) | $s = 2\pi r$, $A = \pi r^2$ | $s = 2\pi r$, $A = \pi r^2$ |
Both sides agree on the answer (of course), but only one side makes the formulas easy to remember.
7. Common pitfalls
Calculators and programming languages each have a mode (or function) for degrees vs. radians. sin(30) in radian mode is roughly $-0.988$ — wildly different from $\sin(30°) = 0.5$. If a trig answer looks bizarre, the first thing to check is the mode. Always.
$\pi$ is the irrational number $3.14159...$. $\pi$ radians is the angle equal to $180°$. They're related but they're not the same thing — one is a pure number, the other is an angle whose measure happens to be that number. When someone writes "$\theta = \pi$", they mean $\theta$ is an angle of half a turn, not that $\theta$ has the numeric value $3.14$ degrees.
If you measure $\theta$ in degrees but plug it into $s = r\theta$, you'll get an arc length that's off by a factor of $\pi/180$ — about $57$ times too big. Either convert first or use the degree-flavored version of the formula. Don't let the units fight each other.
The clean formula $s = r\theta$ — the one that makes radians look magical — is only true when $\theta$ is in radians. The same is true of $A = \tfrac{1}{2}r^2\theta$. If you ever see one of these formulas applied with a degree value plugged straight in, the answer is wrong by a factor of $\pi/180$. This is one of the most common silent errors in physics and engineering homework.
8. Worked examples
Try each one before opening the solution. The goal is to see whether your steps match the canonical recipe — not just to land on the same final number.
Example 1 · Convert $135°$ to radians
Step 1. Multiply by $\pi/180$:
$$ 135° \cdot \frac{\pi}{180} = \frac{135\pi}{180} $$Step 2. Simplify the fraction. Both numerator and denominator share a factor of $45$:
$$ \frac{135\pi}{180} = \frac{3\pi}{4} $$So $135° = \tfrac{3\pi}{4}$ rad $\approx 2.356$ rad.
Example 2 · Convert $\tfrac{5\pi}{6}$ radians to degrees
Step 1. Multiply by $180/\pi$:
$$ \frac{5\pi}{6} \cdot \frac{180}{\pi} = \frac{5 \cdot 180}{6} $$Step 2. The $\pi$'s cancel and the arithmetic is clean:
$$ \frac{900}{6} = 150 $$So $\tfrac{5\pi}{6}$ rad $= 150°$.
Example 3 · Arc length of a $60°$ slice on a radius-$10$ circle
Step 1. Convert to radians first:
$$ \theta = 60° \cdot \frac{\pi}{180} = \frac{\pi}{3} \text{ rad} $$Step 2. Apply $s = r\theta$:
$$ s = 10 \cdot \frac{\pi}{3} = \frac{10\pi}{3} \approx 10.47 $$The arc is about $10.47$ units long — roughly the same as the radius itself, which makes sense because $60° \approx 1.05$ rad is close to one full radian.
Example 4 · Sector area for $\theta = \tfrac{\pi}{4}$ rad on a radius-$6$ circle
Step 1. Use $A = \tfrac{1}{2} r^2 \theta$ directly — the angle is already in radians:
$$ A = \frac{1}{2} \cdot 6^2 \cdot \frac{\pi}{4} $$Step 2. Compute:
$$ A = \frac{1}{2} \cdot 36 \cdot \frac{\pi}{4} = \frac{36\pi}{8} = \frac{9\pi}{2} \approx 14.14 $$So a one-eighth wedge of a radius-$6$ pie has area about $14.14$ square units.
Example 5 · Evaluate $\sin\!\left(\tfrac{\pi}{6}\right)$ exactly
Step 1. Recognize the angle. $\tfrac{\pi}{6}$ rad $= 30°$ — one of the special-triangle angles.
Step 2. From the $30$–$60$–$90$ triangle, $\sin 30° = \tfrac{1}{2}$.
$$ \sin\!\left(\frac{\pi}{6}\right) = \frac{1}{2} $$The point of the example: a "scary"-looking expression in radians often reduces to a very familiar number once you translate the angle. Memorizing the common conversions makes special-angle problems essentially instant.