Everyone learns that $180^\circ = \pi$ radians and then memorises it forever without knowing why. The reason is a picture: take the radius of a circle, bend it around the rim, and the angle you have just swept is one radian.

Concept What a radian means

A radian is the angle you get when you travel a distance equal to the radius along the circumference. Lay the radius flat against the rim starting from a point; the arc it covers subtends one radian at the centre, which works out to about

57.3^\circ

Concept Why 180° is exactly π

Keep laying the radius around the rim and count how many it takes. To reach the halfway point, $180^\circ$ , you need $\pi \approx 3.14$ radius lengths. To go all the way round you need $2\pi \approx 6.28$ .

180^\circ = \pi

radians

360^\circ = 2\pi

radians

This is not a coincidence with the circumference formula $C = 2\pi r$ . It is the same fact: the rim is $2\pi$ radius lengths long, so a full turn is $2\pi$ radians.

Note The four to memorise

Split the circle into its four quadrants and learn the boundary angles once:

90^\circ = \dfrac{\pi}{2}

180^\circ = \pi

270^\circ = \dfrac{3\pi}{2}

360^\circ = 2\pi

Example Halve and halve again

For the angles between the boundaries, split the ones you already know.

45^\circ

is half of

90^\circ

, so

\dfrac{\pi}{2} \div 2 = \dfrac{\pi}{4}

135^\circ = 90^\circ + 45^\circ

, so

\dfrac{\pi}{2} + \dfrac{\pi}{4} = \dfrac{3\pi}{4}

The same pattern of adding $\dfrac{\pi}{4}$ continues around the rest of the circle.

Example Any angle at all

For an angle like $30^\circ$ or $10^\circ$ that no halving lands on, use the general rule: multiply by $\pi$ and divide by $180$ .

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

30^\circ \times \dfrac{\pi}{180} = \dfrac{\pi}{6}

20^\circ \times \dfrac{\pi}{180} = \dfrac{\pi}{9}

10^\circ \times \dfrac{\pi}{180} = \dfrac{\pi}{18}

Summary

A radian is the angle whose arc equals the radius, about $57.3^\circ$ .
It takes $\pi \approx 3.14$ radius lengths to reach $180^\circ$ , so $180^\circ = \pi$ .
A full turn is $2\pi$ , the same $2\pi$ as in $C = 2\pi r$ .
Memorise the quadrant boundaries: $\tfrac{\pi}{2},\ \pi,\ \tfrac{3\pi}{2},\ 2\pi$ .
Halve and add $\tfrac{\pi}{4}$ for the angles in between: $45^\circ = \tfrac{\pi}{4}$ , $135^\circ = \tfrac{3\pi}{4}$ .
For any other angle, multiply by $\tfrac{\pi}{180}$ .

What Is a Radian