How to convert between degrees, radians, gradians and turns
An angle measures rotation, and the same rotation can be written in several units. A full circle is 360 degrees, 2π radians, 400 gradians, or 1 turn. Once you know that, every conversion is a single multiplication.
The most common conversion is degrees to radians. Because 360° equals 2π radians, 1° equals π/180 radians, which gives the formula radians = degrees × π/180. Going the other way, degrees = radians × 180/π. Gradians follow from gradians = degrees × 10/9 (a right angle is 90° or 100 gon), and turns = degrees / 360.
Worked example: convert 180° to radians. radians = 180 × π/180 = π ≈ 3.1416. In gradians, 180 × 10/9 = 200 gon, and in turns, 180 / 360 = 0.5 of a full revolution.
Common angles reference table
| Degrees | Radians (π form) | Radians (decimal) | Gradians |
|---|---|---|---|
| 0° | 0 | 0.0000 | 0 |
| 30° | π/6 | 0.5236 | 33.33 |
| 45° | π/4 | 0.7854 | 50 |
| 60° | π/3 | 1.0472 | 66.67 |
| 90° | π/2 | 1.5708 | 100 |
| 180° | π | 3.1416 | 200 |
| 270° | 3π/2 | 4.7124 | 300 |
| 360° | 2π | 6.2832 | 400 |
Reference note: results are rounded for display. The "in terms of π" form is found by simplifying the fraction degrees/180, so 90° → 90/180 = 1/2 → π/2.
Frequently asked questions
- How do I convert degrees to radians?
- Multiply the angle in degrees by π/180. For example, 90 × π/180 = π/2 ≈ 1.5708 radians. The factor π/180 comes from a full circle being both 360° and 2π radians.
- What is 90 degrees in radians?
- 90 degrees equals π/2 radians, which is approximately 1.5708. You get this by multiplying 90 by π/180, which simplifies to π/2.
- How do I convert radians to degrees?
- Multiply the angle in radians by 180/π. For example, π radians × 180/π = 180 degrees, and 1 radian × 180/π ≈ 57.2958 degrees.
- What is a radian?
- A radian is the angle made at the centre of a circle by an arc whose length equals the circle's radius. A full circle is 2π radians, so one radian is about 57.2958 degrees.
- What is a gradian?
- A gradian (also called a gon or grade) divides a right angle into 100 parts, so a full circle is 400 gradians. To convert degrees to gradians, multiply by 10/9; 90 degrees equals 100 gradians.
- What is 360 degrees in radians?
- 360 degrees equals 2π radians, which is approximately 6.2832. A full turn is 360°, 2π radians, 400 gradians, or 1 revolution.