You need to know how to convert degrees to radians and radians to degrees in geometry, trigonometry, and science. The radian is the SI derived unit of measuring angles. It is a dimensionless value or has the unit symbol r or rad. Meanwhile, the degree (°) measures a plane angle where a complete rotation is 360°. The degree is easy to measure and finds practical use in astronomy and geography.
2π = 360°
π = 180°
Here are the degrees to radians and radians degrees formulas and examples showing how to work the angle conversions.
Degrees to Radians Formula
The formula for converting degrees to radians is:
radians = degrees x π/180
Note that radian measurements retain the π. You can report radian measurements as decimal values by approximately pi as 3.14, but this is not common.
How to Convert Degrees to Radians
Here are the step for converting degrees to radians:
- Write the angle in degrees.
- Multiply it by π/180.
- Retain the π and simplify the fraction.
If you need the radian measurement in decimal format, remember the value of pi is approximately 3.14 or 22/7.
Degrees to Radians Example
For example, convert 90° to radians.
radians = degrees x π/180
radians = 90 x π/180 = π/2
Radians to Degrees Formula
The formula for converting radians to degrees is:
degrees = radians x 180/π
Degrees are either whole numbers or decimal numbers. But, you still want to keep the π in the radian value (rather than converting it to a decimal) so it will cancel out in the calculation.
How to Convert Radians to Degrees
Here are the steps for converting radians to degrees:
- Write the radian value.
- Multiply this value by 180 and divide it by π.
If the radian value contains π, then it will cancel out of the formula and all you really need to do is multiply the fraction by 180 to get the answer!
Radians to Degrees Example
For example, convert π/6 to degrees.
degrees = radians x 180/π = π/6 x 180/π = 180/6 = 30°.
Negative Degrees or Radians
The unit conversions work fine with negative degrees or radians.
For example, convert -180° to radians.
radians = degrees x π/180 = -180 x π/180 = -π
For example, convert -3π/2 radians to degrees.
degrees = radians x 180/π = -3π/2 x 180/π = -1.5 x 180 = -270°
Table of Degree and Radian Conversions
This table covers the most common degrees to radians conversions:
| Angle in Degrees | Angle in Radians |
|---|---|
| 0° | 0 rad |
| 30° | π/6 ≈ 0.524 rad |
| 45° | π/4 ≈ 0.785 rad |
| 60° | π/3 ≈ 1.047 rad |
| 90° | π/2 ≈ 1.571 rad |
| 120° | 2π/3 ≈ 2.094 rad |
| 150° | 5π/6 ≈ 2.618 rad |
| 180° | π ≈ 3.14 rad |
| 210° | 7π/6 ≈ 3.665 rad |
| 270° | 3π/2 ≈ 4.713 rad |
| 360° | 2π ≈ 6.283 rad |
What Is 1 Degree or 1 Radian?
- One degree is π/180, which is approximately equal to 0.0174533 radians.
- One radian is approximately 57.2958°.
References
- Cajori, Florian (1929). History of Mathematical Notations. 2. Dover Publications. ISBN 0-486-67766-4.
- Mohr, J. C.; Phillips, W. D. (2015). “Dimensionless Units in the SI”. Metrologia. 52 (1): 40–47. doi:10.1088/0026-1394/52/1/40
- Muir, Thos. (1910). “The Term “Radian” in Trigonometry”. Nature. 83 (2110): 156. doi:10.1038/083156a0
- Protter, Murray H.; Morrey, Charles B., Jr. (1970). College Calculus with Analytic Geometry (2nd ed.). Reading: Addison-Wesley.