Calculate radians from degrees, or degrees from radians, with the exact pi form and step-by-step results.
Degrees to Radians Formula
A full circle is 360 degrees or 2 pi radians, so 180 degrees equals pi radians. That single fact gives you both conversion formulas.
radians = degrees * pi / 180
degrees = radians * 180 / pi
- degrees = the angle measured in degrees
- radians = the angle measured in radians
- pi = the constant 3.14159..., the ratio of a circle's circumference to its diameter
To go from degrees to radians you multiply by pi and divide by 180. To go the other way you multiply the radian value by 180 and divide by pi. The calculator runs whichever direction you pick in the Convert menu. In degrees to radians mode it also returns the exact answer as a multiple of pi, such as pi/2 for 90 degrees, by reducing the fraction degrees/180 to lowest terms. The advanced options let you set how many decimal places to display and add gradians and turns, where one turn is a full circle and 400 gradians also make a full circle.
Common Angle Conversions
These are the angles you will meet most often. The exact column shows the radian value written as a multiple of pi, and the decimal column rounds to four places.
| Degrees | Radians (exact) | Radians (decimal) |
|---|---|---|
| 0 | 0 | 0.0000 |
| 30 | pi/6 | 0.5236 |
| 45 | pi/4 | 0.7854 |
| 60 | pi/3 | 1.0472 |
| 90 | pi/2 | 1.5708 |
| 120 | 2pi/3 | 2.0944 |
| 180 | pi | 3.1416 |
| 270 | 3pi/2 | 4.7124 |
| 360 | 2pi | 6.2832 |
One radian is about 57.2958 degrees, which is the value you get when you set the radian input to 1.
| Radians | Degrees |
|---|---|
| 1 | 57.2958 |
| pi/2 | 90 |
| pi | 180 |
| 2pi | 360 |
Example Problems
Example 1: Convert 90 degrees to radians. Multiply 90 by pi and divide by 180, which gives 90 pi / 180. The fraction 90/180 reduces to 1/2, so the exact answer is pi/2, and as a decimal that is about 1.5708 radians.
Example 2: Convert 2.5 radians to degrees. Multiply 2.5 by 180 and divide by pi, which gives 450 / pi, or about 143.2394 degrees.
Frequently Asked Questions
Why is 180 degrees equal to pi radians? A radian is defined so that the arc length around a circle equals the radius times the angle in radians. The full circumference is 2 pi times the radius, so a full turn of 360 degrees is 2 pi radians. Half of that, 180 degrees, is therefore pi radians, and this relationship is what every degree and radian conversion is built on.
How do I enter an answer that involves pi? In degrees to radians mode the calculator shows the exact result as a multiple of pi for you, so you do not need to type pi yourself. In radians to degrees mode you can tick the box that says the value is a multiple of pi and enter just the coefficient. For example, to convert pi/2 radians you tick the box and enter 0.5.
When should I use radians instead of degrees? Degrees are common in geometry, navigation, and everyday angle measurement. Radians are the standard in calculus, physics, and most programming languages, where trigonometric functions expect their input in radians. Convert to radians whenever a formula or function assumes that unit.
