Degrees to Radians
To convert degrees to radians, multiply by π and divide by 180. Here is why that works, how to get the exact π form, and a chart of the angles you will use most.
radians = degrees × π / 180 · degrees = radians × 180 / π
Need a quick answer? The angle converter converts instantly between degrees, radians, gradians and DMS.
What is a radian?
One radian is the angle you get when you wrap the radius of a circle along its edge: the arc is exactly as long as the radius. Because the circumference of a circle is 2π times its radius, a full turn is 2π radians. So 360° = 2π rad, and dividing both sides by 2 gives the key fact: 180° = π radians. Every conversion comes from that.
How to convert degrees to radians (step by step)
- Write down the angle in degrees. For example 135°.
- Multiply by π/180. 135 × π/180 = 135π/180.
- Simplify the fraction. Divide top and bottom by their greatest common factor (45): 135π/180 = 3π/4.
- Get a decimal if you need one. 3π/4 ≈ 3 × 3.14159 ÷ 4 ≈ 2.3562 radians.
Degrees to radians chart
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0° | 0 | 0 |
| 15° | π/12 | 0.2618 |
| 30° | π/6 | 0.5236 |
| 45° | π/4 | 0.7854 |
| 60° | π/3 | 1.0472 |
| 75° | 5π/12 | 1.309 |
| 90° | π/2 | 1.5708 |
| 120° | 2π/3 | 2.0944 |
| 135° | 3π/4 | 2.3562 |
| 150° | 5π/6 | 2.618 |
| 180° | π | 3.1416 |
| 210° | 7π/6 | 3.6652 |
| 225° | 5π/4 | 3.927 |
| 240° | 4π/3 | 4.1888 |
| 270° | 3π/2 | 4.7124 |
| 300° | 5π/3 | 5.236 |
| 315° | 7π/4 | 5.4978 |
| 330° | 11π/6 | 5.7596 |
| 360° | 2π | 6.2832 |
Radians to degrees
Go the other way by multiplying by 180/π. With an exact π value, the π cancels: 5π/6 × 180/π = 5 × 180 / 6 = 150°. With a decimal, divide by π: 2 rad × 180/π ≈ 114.59°.
More examples
- 1° = π/180 ≈ 0.01745 rad
- 22.5° = π/8 ≈ 0.3927 rad
- 100° = 5π/9 ≈ 1.7453 rad
- 270° = 3π/2 ≈ 4.7124 rad
- 57.2958° ≈ 1 rad
Common mistakes
- Calculator in the wrong mode. Check for DEG or RAD on the display before using sin, cos or tan.
- Forgetting to simplify. 90π/180 is correct but should be written π/2.
- Mixing units in code. Most programming languages expect radians. Convert with
deg * Math.PI / 180. - Confusing radians with gradians. "GRAD" on a calculator means gradians (400 per circle), not radians.
Frequently asked questions
What is the formula to convert degrees to radians?
radians = degrees × π / 180. For example, 60° × π/180 = π/3 ≈ 1.0472 rad.
How many degrees is 1 radian?
1 radian = 180/π ≈ 57.2958°, or about 57° 17′ 45″.
Why do we use radians?
Radians make the math simpler: the arc length of a circle is just s = r × θ, and in calculus the derivative of sin x is cos x only when x is in radians. That is why programming languages (JavaScript Math.sin, Python math.sin, Excel SIN) expect radians.
Why does my calculator give the wrong answer for sin(30)?
It is probably in radian mode, so it calculates the sine of 30 radians (−0.988) instead of 30° (0.5). Switch it to DEG mode, or convert first: 30° = π/6.