It is the central angle that cuts an arc as long as the radius. The definition appeals to no convention about splitting a turn into parts, which is why the radian is called the natural measure of angle — and the result does not depend on how large the circle is.

How many degrees is one radian?

About 57.3°, exactly 180°/π. The number is irrational, so it is never written out exactly in degrees — which is why angles in radians are written as multiples of π, for example π/6 rather than 0.5236.

How many radians are in a full turn?

Exactly 2π, about 6.283. It follows from the circumference: it is 2πr long, and each radian corresponds to an arc of length r, so the whole circle holds 2π of them. That also gives 180° = π rad.

How do I convert degrees to radians?

Multiply by π/180°. For instance 60° = 60 · π/180 = π/3. The other way multiply by 180°/π: 3π/4 = 3 · 180°/4 = 135°. A sanity check: converting to radians must make the number smaller (60 → about 1.05), converting to degrees must make it larger.

How do I find an arc length in radian measure?

From l = rα, with α in radians. For r = 6 cm and α = π/3 that is l = 6 · π/3 = 2π ≈ 6.28 cm. It is the same arc that degrees compute as 2πr · α/360° — radian measure removes the division by 360 from the calculation.

How do I find the area of a sector in radians?

From P = ½r²α. For r = 6 cm and α = π/3 that is P = ½ · 36 · π/3 = 6π ≈ 18.85 cm². A check: for a full turn, α = 2π, the formula gives ½r² · 2π = πr², the area of the whole disc.

Is the radian a unit?

Formally no — it is a ratio of two lengths (arc over radius), so it is dimensionless. The "rad" abbreviation is written only to make clear that degrees are not meant. That is why l = rα carries no conversion factor and the answer inherits the unit of the radius.

Why do programming libraries take trigonometric arguments in radians?

Because the whole of calculus, which those libraries rest on, is built in radian measure: the derivative of the sine is the cosine only when the argument is in radians, and for small angles sin x ≈ x. In degrees both facts would carry an extra factor of π/180.

Advanced level

Radian measure

A radian is the angle that cuts an arc equal to the radius. See where the 2π of a full turn comes from, how to convert degrees and radians both ways, and why the arc-length and sector-area formulas collapse to a single multiplication in this measure.

Before you start

This topic builds on earlier ideas. Before you start, it's worth working through the lessons below — they'll make everything click:

The unit circleThe unit circle carries the sine and the cosine from a triangle onto the whole circle. See how the coordinates of a point become the values of the functions, what signs they take in the four quadrants, and what a negative angle or one past a full turn means.
The circle: angles, tangent, regular polygonsAn inscribed angle is always half the central angle on the same arc — and from that one sentence follow the right angle over a diameter, the construction of a tangent and half of every exam question about circles. Plus chords, tangent segments, regular polygons, and the length of an arc and the area of a sector.

Where this is used

Real situations where you count exactly the way this lesson teaches:

The socket in the wall
Mains voltage runs as a sine, u(t) = 325 · sin(ωt), where ω = 2π · 50 Hz ≈ 314 rad/s. Four milliseconds after a zero crossing the angle is 314 · 0.004 = 1.256 rad and the voltage is 325 · sin(1.256) ≈ 309 V. The calculator has to be in RAD mode for that: in DEG it computes sin 1.256° ≈ 0.022 and answers 7 V instead of 309 V.
A bike computer
A wheel of radius 0.35 m turns through 2 rad. The bike moves l = 0.35 · 2 = 0.70 m — one multiplication, no dividing by 360. That is exactly what a bike computer does: it counts revolutions, multiplies by 2π · 0.35 ≈ 2.20 m and shows the distance ridden.
Robotics
Servo controllers take angles in radians, because angular velocity and torque are expressed in rad/s. An arm 0.4 m long rotated by 0.5 rad moves the gripper 0.4 · 0.5 = 0.20 m along the arc. Sending the controller 30 instead of π/6 ≈ 0.524 produces a motion almost 60 times too large — that is how many degrees one radian holds.
Convert it here: Angle
Optics and reticles
The milliradian (1 mrad = 0.001 rad) is the correction unit on rifle scopes and theodolites for exactly one reason: l = rα. At 300 m one mrad spans 300 · 0.001 = 0.30 m, so "two clicks of 0.1 mrad" shifts the point of impact by 6 cm with no trigonometry involved.
Cutting sheet metal
A sheet sector of radius 40 cm and angle 1.2 rad has an area of ½ · 40² · 1.2 = 960 cm² and a curved edge 40 · 1.2 = 48 cm long. Two multiplications are enough to price the material and order the edging tape — in degrees the same numbers need a division by 360 first.

All formulas

One radian
$1\ \text{rad} = \frac{180^\circ}{\pi} \approx 57.30^\circ$
the angle whose arc is as long as the radius
A full turn and a half turn
$360^\circ = 2\pi\ \text{rad}, \qquad 180^\circ = \pi\ \text{rad}$
a circumference of 2πr holds 2π radii
Degrees → radians
$\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}$
the result is a smaller number than the degree count
Radians → degrees
$\alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}$
π cancels and the degree count is left
Arc length
$l = r \alpha$
α in radians — no conversion factor at all
Sector area
$P = \frac{1}{2} r^2 \alpha$
α in radians; α = 2π gives πr²

Splitting a full turn into $360$ parts is a convention — a convenient one, because 360 divides by a great many numbers, but one inherited from the Babylonian base-sixty system. The circle itself knows nothing about it. The measure the circle supplies on its own looks different.

An angle measured by its arc

Take a circle of radius $r$ and mark off an arc exactly $r$ long. The central angle standing on that arc is one radian.

One radian: arc AB is the same length as radius SA. The angle comes out the same in every circle, because the arc is measured in RADII.

What matters is that the definition does not depend on the size of the circle. Double the radius and the arc of one radius doubles with it — while the angle between the arms stays put. Radian measure is therefore a ratio:

\alpha = \frac{l}{r}

that is, the number of radii that fit into the arc. Two lengths cancel, so the radian is dimensionless; the abbreviation $\text{rad}$ is written only to tell it apart from degrees.

Where the 2π comes from

The whole circle is $2\pi r$ long. How many arcs of length $r$ fit into it?

\frac{2\pi r}{r} = 2\pi

So a full turn is $2\pi$ radians, and a half turn is half of that:

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

Everything else follows from that single equality. One radian is

1\ \text{rad} = \frac{180^\circ}{\pi} \approx 57.30^\circ

The number is irrational, so it never gets written out exactly in degrees. That is why angles in radian measure are written as multiples of $\pi$ — $\frac{\pi}{6}$ , $\frac{3\pi}{4}$ , $2\pi$ — rather than as decimal expansions.

Converting both ways

Since $180^\circ = \pi$ , the conversion factor each way is a fraction that equals one:

\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}, \qquad \alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}

degrees	$30^\circ$	$45^\circ$	$60^\circ$	$90^\circ$	$120^\circ$	$180^\circ$	$270^\circ$	$360^\circ$
radians	$\frac{\pi}{6}$	$\frac{\pi}{4}$	$\frac{\pi}{3}$	$\frac{\pi}{2}$	$\frac{2\pi}{3}$	$\pi$	$\frac{3\pi}{2}$	$2\pi$

Convert 135° to radians, and 5π/6 to degrees.

If you need an actual value converted — gradians, arcminutes and arcseconds included — the angle converter will do it.

Arc length

Now it is clear what this measure was invented for. The definition $\alpha = \frac{l}{r}$ , rearranged for $l$ , gives a formula with no conversion factor at all:

l = r \alpha

A sector of angle α: the arc is rα long and the shaded region is ½r²α. Both formulas need the angle in radians.

Compare it with the degree version you know from the circle:

l = 2\pi r \cdot \frac{\alpha_{\deg}}{360^\circ}

Both give the same number — but the second one has to say first what fraction of a full turn the angle is. Radian measure has that fraction built into the number itself.

Find the length of an arc of radius 6 cm subtending a central angle of π/3.

Sector area

The same calculation for area. A sector of angle $\alpha$ is the same fraction of the disc that $\alpha$ is of the full turn $2\pi$ :

P = \pi r^2 \cdot \frac{\alpha}{2\pi} = \frac{1}{2} r^2 \alpha

An extreme check: for $\alpha = 2\pi$ the formula gives $\frac{1}{2} r^2 \cdot 2\pi = \pi r^2$ , the area of the whole disc ✓.

A sector has radius 10 cm and angle 3π/4. Find its area and the length of its arc.

Why this is the natural measure

Three facts that simply do not hold in degrees.

The arc and sector formulas carry no conversion factor. $l = r\alpha$ and $P = \frac{1}{2}r^2\alpha$ are one multiplication each. In degrees both need an extra division by $360$ .
For small angles $\sin \alpha \approx \alpha$ . At $\alpha = 0.05$ rad we have $\sin \alpha = 0.049979\ldots$ — a difference in the fifth decimal place. That approximation underpins calculations in optics, mechanics and navigation, and in degrees it would be false ( $\sin 0.05^\circ \approx 0.00087$ ).
The argument axis of a trigonometric graph is scaled in $\pi$ . The graph of $y = \sin x$ has period $2\pi$ precisely because its argument is a radian measure; in degrees the period would be $360$ and the curve would be stretched two hundredfold against its value axis.

Hence the practical rule: calculators and programming libraries work in radians. Math.sin(1) in JavaScript is the sine of one radian, about $0.841$ , not the sine of one degree.

Exercises

In the conversion questions the answer is either a multiple of π (e.g. π/3, 3π/4) or a plain degree count (e.g. 60). In the arc and sector questions write the exact value — again as a multiple of π, e.g. 2π or 15π/2.

Practice

Work through a set of exercises — they get harder as you go. At the end you'll see your score and the mistakes worth reviewing.

Exercise 1 of 8Score: 0

45° → rad

Common mistakes

Multiplying by $\frac{180}{\pi}$ instead of $\frac{\pi}{180}$ — converting degrees into radians must produce a smaller number.
Treating $\pi$ as a unit — $\frac{\pi}{3}$ is an ordinary number (about $1.05$ ), not "π thirds of a degree".
Using $l = r\alpha$ with the angle in degrees — for $r = 6$ and $\alpha = 60^\circ$ it returns $360$ instead of $6.28$ . The angle has to be converted first.
A calculator in DEG mode with an argument in radians — quick test: $\sin 1$ must give $0.841$ . If it gives $0.0175$ , the calculator is working in degrees.
Confusing $\frac{1}{2}r^2\alpha$ with $r^2\alpha$ — the half is not decoration; without it the area comes out twice too large.
Rounding $\pi$ mid-calculation — cancel the fractions first and substitute $\pi \approx 3.14$ only at the end, and only if you have to.

Formula card

Topic: Radian measure

One radian
$1\ \text{rad} = \frac{180^\circ}{\pi} \approx 57.30^\circ$
the angle whose arc is as long as the radius
A full turn and a half turn
$360^\circ = 2\pi\ \text{rad}, \qquad 180^\circ = \pi\ \text{rad}$
a circumference of 2πr holds 2π radii
Degrees → radians
$\alpha_{\text{rad}} = \alpha_{\deg} \cdot \frac{\pi}{180^\circ}$
the result is a smaller number than the degree count
Radians → degrees
$\alpha_{\deg} = \alpha_{\text{rad}} \cdot \frac{180^\circ}{\pi}$
π cancels and the degree count is left
Arc length
$l = r \alpha$
α in radians — no conversion factor at all
Sector area
$P = \frac{1}{2} r^2 \alpha$
α in radians; α = 2π gives πr²

One radian: the central angle whose arc AB is exactly as long as the radius. It does not depend on how large the circle is.

A sector of angle α: the arc is rα long and the sector area is ½r²α. Both formulas hold only when α is measured in radians.

Frequently asked questions

A luminous unit circle with a radius sweeping out an angle, a sine wave unrolling from it as a streak of light, beside a right triangle drawn in stars, against a cosmic nebula in violet and magenta

Trigonometry

Why sine doesn't depend on the size of the triangle

The height of a tower, the width of a river, the distance to a ship offshore — all of it can be computed without going anywhere near the thing being measured. The whole of trigonometry rests on one observation: the ratio of two sides in a right triangle depends only on the angle, not on how big the triangle is. That is what turns an angle into a number — and lets it be tabulated once and for all.

13 min