Degrees and Radians

What a radian actually is

A degree is an arbitrary unit: 360 of them make a full circle, likely borrowed from the 360 days of an ancient calendar year. A radian, the SI unit for angles, is defined geometrically instead: it's the angle you sweep out when the arc length traveled equals the radius:

\text{Radian} = \dfrac{\text{arc length}}{\text{radius}}

A full circle's circumference is $2\pi r$ (exactly $2\pi$ radius-lengths around), so a full $360^\circ$ turn is exactly $2\pi$ radians.

Converting between the two

Since $180^\circ = \pi$ radians, the two conversion factors are:

\text{radians} = \text{degrees} \times \dfrac{\pi}{180}, \qquad \text{degrees} = \text{radians} \times \dfrac{180}{\pi}

One radian is $\dfrac{180^\circ}{\pi} \approx 57.3^\circ$ , a little under a sixth of a full turn.

Standard-angle conversions

0^\circ=0, \quad 30^\circ=\dfrac{\pi}{6}, \quad 45^\circ=\dfrac{\pi}{4}, \quad 60^\circ=\dfrac{\pi}{3}, \quad 90^\circ=\dfrac{\pi}{2}, \quad 180^\circ=\pi

Larger angles worth knowing by heart

Degrees	Radians
$120^\circ$	$\dfrac{2\pi}{3}$
$135^\circ$	$\dfrac{3\pi}{4}$
$150^\circ$	$\dfrac{5\pi}{6}$
$210^\circ$	$\dfrac{7\pi}{6}$
$270^\circ$	$\dfrac{3\pi}{2}$
$360^\circ$	$2\pi$

Worked example: degrees to radians

Convert $150^\circ$ to radians.

Solution: Multiply by $\tfrac{\pi}{180}$ :

150 \times \dfrac{\pi}{180} = \dfrac{5\pi}{6}

Worked example: radians to degrees

Convert $\dfrac{5\pi}{4}$ radians to degrees.

Solution: Multiply by $\tfrac{180}{\pi}$ :

\dfrac{5\pi}{4} \times \dfrac{180}{\pi} = 5 \times 45 = 225^\circ

Worked example: degrees and minutes

Convert $40^\circ 20'$ to radians.

Solution: There are 60 minutes in a degree, so first write the angle in degrees alone:

40^\circ 20' = 40\tfrac{1}{3}^\circ = \dfrac{121}{3}^\circ

\dfrac{121}{3} \times \dfrac{\pi}{180} = \dfrac{121\pi}{540}

Arc length and sector area

Radians make these two formulas as simple as they can be. For a circle of radius $r$ and a central angle $\theta$ in radians:

\text{arc length } s = r\theta, \qquad \text{sector area } A = \dfrac{1}{2}r^2\theta

In degrees the same formulas carry an extra $\dfrac{\pi}{180}$ , which is exactly the factor radians absorb.

Worked example: a clock's minute hand

The minute hand of a clock is 21 cm long. How far does its tip travel in 10 minutes? (Take $\pi = \dfrac{22}{7}$ .)

Solution: The hand turns $360^\circ$ in 60 minutes, so $60^\circ$ in 10 minutes, which is $\dfrac{\pi}{3}$ radians:

s = r\theta = 21 \times \dfrac{\pi}{3} = 7\pi = 7 \times \dfrac{22}{7} = 22 \text{ cm}

Common mistakes

Using degrees in $s = r\theta$ . With $\theta = 60$ instead of $\dfrac{\pi}{3}$ , the clock answer comes out as 1,260 cm.
Leaving the calculator in the wrong mode. $\sin 30$ is 0.5 in degree mode and about $-0.988$ in radian mode. Check the mode before any trig question.
Forgetting to convert minutes first. $40^\circ 20'$ is not $40.2^\circ$ . Twenty minutes is a third of a degree.
Thinking radians need a $\pi$ . $\dfrac{\pi}{3}$ is a radian measure, but so is 2. An angle of 2 radians is about $114.6^\circ$ .

Where this comes up in exams

The 2025-26 CBSE Class XI syllabus covers measuring angles in radians and in degrees and converting between the two, at the start of its trigonometric functions unit.

Checked against CBSE Mathematics curriculum, Classes XI-XII (2025-26). Syllabi can change from year to year, so confirm with the latest official notification for your exam.

Fit it into your week with the free timetable generator.
Mark it off as you revise in the syllabus tracker.
See how long you have left: exam countdowns.