Until now, you have probably always measured angles in degrees (0° to 360°). In the exact sciences, however, we use the radian. This unit makes calculations with waves and circular motions much simpler.

1. What is a radian?

One radian is the angle at which the arc length of the circle is exactly equal to the radius.

The logic of the circle:

The circumference of a circle is 2π · r.
In the unit circle (r = 1), the full circumference is therefore exactly 2π.
This means that 360° is equal to 2π radians.

2. Quick Conversion

The most important ratio to remember is the semi-circle:

180° = π radians

From degrees to radians:
Multiply by π180
Example: 60° → 60 · π 180 = 1 3 π

From radians to degrees:
Multiply by 180π
Example: 1 2 π → π 2 · 180 π = 90°

3. Standard Angles

Make sure you can dream these values; they recur in almost every trigonometry problem:

Degrees	Radians	Description
0°	0	Starting point
30°	16π	Sixth of a circle
45°	14π	Quarter of 180°
90°	12π	Quarter circle
180°	π	Semi-circle
360°	2π	Full circle