What Is a Radian? Converting Between Degrees and Radians

What a radian actually is, why a full turn is $2\pi$ , how to convert between degrees and radians, the common angles to know, and why calculus insists on radians.

3 min read Supports 3 lessons

Degrees split a full turn into $360$ parts, a number chosen thousands of years ago for convenience. Radians measure angles using the circle itself.

Take the radius of a circle and bend it around the edge. The angle it marks out at the centre is one radian, a little over $57°$ .

Why a full turn is 2π

The whole circumference is $2\pi r$ , so exactly $2\pi$ radius-lengths fit around the circle. A full turn is therefore $2\pi$ radians, and a half turn is $\pi$ .

Conversion

180^\circ = \pi \text{ rad} \qquad \text{deg} \to \text{rad}: \times\frac{\pi}{180} \qquad \text{rad} \to \text{deg}: \times\frac{180}{\pi}

$135^\circ$
Convert to radians.

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Angles to know

30^\circ = \tfrac{\pi}{6} \quad 45^\circ = \tfrac{\pi}{4} \quad 60^\circ = \tfrac{\pi}{3} \quad 90^\circ = \tfrac{\pi}{2} \quad 180^\circ = \pi \quad 360^\circ = 2\pi

Tip

Think in fractions of a half turn. $\frac{\pi}{4}$ is a quarter of a half turn, $45°$ . $\frac{5\pi}{6}$ is five sixths of a half turn, $150°$ .

Drag the point. Both degrees and radians are shown.

Arcs become simple

In radians, the length of an arc is just the radius times the angle, and a sector's area is $\frac12 r^2\theta$ . No $360$ in sight.

\text{arc length} = r\theta \qquad \text{sector area} = \tfrac12 r^2\theta

Why calculus uses radians

Key idea

The derivative of $\sin x$ is $\cos x$ only when $x$ is in radians. In degrees an awkward factor of $\frac{\pi}{180}$ appears in every formula. Radians are the unit in which the maths comes out clean.

Near zero, $\sin x \approx x$ when $x$ is in radians: $\sin(0.1) \approx 0.0998$ . This small fact is behind the whole of trigonometric calculus.

Watch out

Check your calculator's mode. $\sin(30)$ in radian mode is about $-0.988$ , not $0.5$ .

Common questions

How many degrees is one radian?: One radian is $\frac{180}{\pi} \approx 57.3°$ .
Why do we use radians instead of degrees?: Radians are based on the circle itself, so arc lengths, areas and calculus formulas come out without conversion factors. The derivative of $\sin x$ is $\cos x$ only in radians.

Practise radians

The Radians lesson has worked examples and unlimited practice that adapts to you.

Open the lesson

Common questions

How many degrees is one radian?

One radian is $\frac{180}{\pi} \approx 57.3°$ .

Why do we use radians instead of degrees?

Radians are based on the circle itself, so arc lengths, areas and calculus formulas come out without conversion factors. The derivative of $\sin x$ is $\cos x$ only in radians.