Degrees to Radians Converter
Enter any angle in degrees to get its exact value in radians as both a pi-fraction (like pi/4) and a decimal. The converter also returns the sine, cosine, and tangent for that angle, and you can flip the direction to convert radians back to degrees. The reference table below lists every common angle you are likely to meet in class or on the job.
How to convert degrees to radians
The conversion is a single multiplication: multiply the angle in degrees by pi divided by 180. Written as a formula, radians = degrees x (pi / 180). Because pi/180 is approximately 0.017453, you can also multiply by that decimal if an exact fraction is not needed. For example, 90 degrees converts to 90 x (pi/180) = pi/2, which is approximately 1.5708 radians. Going the other way, multiply radians by 180/pi (approximately 57.2958) to recover degrees.
Why radians exist and when to use them
A radian is the angle subtended at the center of a circle by an arc equal in length to the radius. One full circle is exactly 2*pi radians, a half circle is pi radians, and a quarter circle is pi/2 radians. The radian system is not arbitrary: when you use radians, the derivatives of sin and cos are cos and -sin, with no extra scaling factor. That makes radians the natural unit for calculus, physics, and engineering. Degrees are better for navigation, surveying, and everyday geometry where whole numbers feel intuitive. Trigonometric functions in most programming languages (Math.sin, numpy.sin, etc.) expect radians, so the conversion is a routine task for developers.
The unit circle and exact values
The angles 0, 30, 45, 60, and 90 degrees (and their reflections through the other three quadrants) are called the special angles. They produce exact radical values for sin, cos, and tan, and they convert to clean pi-fractions: pi/6, pi/4, pi/3, and pi/2. The reference table above lists all 17 standard angles from 0 to 360 with their exact fractions and trig values. Memorising the pattern - 0, pi/6, pi/4, pi/3, pi/2 for the first quadrant - is the fastest way to handle trig problems without a calculator. For any other angle, multiply by pi/180 to get the decimal radian value.
Converting radians to degrees
To reverse the conversion, multiply radians by 180/pi. For example, pi/3 radians = (pi/3) x (180/pi) = 60 degrees. In decimal form, 1 radian x (180/pi) is approximately 57.2958 degrees. Use the direction selector at the top of the calculator to switch to radians-to-degrees mode and enter any radian value. The calculator handles decimals, negatives, and values beyond 2*pi (i.e. angles that have gone around more than once).
Common angle conversions
| Degrees | Radians (fraction) | Radians (decimal) | sin | cos | tan |
|---|---|---|---|---|---|
| 0 | 0 | 0.000000 | 0 | 1 | 0 |
| 30 | pi/6 | 0.523599 | 1/2 | sqrt(3)/2 | 1/sqrt(3) |
| 45 | pi/4 | 0.785398 | sqrt(2)/2 | sqrt(2)/2 | 1 |
| 60 | pi/3 | 1.047198 | sqrt(3)/2 | 1/2 | sqrt(3) |
| 90 | pi/2 | 1.570796 | 1 | 0 | undefined |
| 120 | 2pi/3 | 2.094395 | sqrt(3)/2 | -1/2 | -sqrt(3) |
| 135 | 3pi/4 | 2.356194 | sqrt(2)/2 | -sqrt(2)/2 | -1 |
| 150 | 5pi/6 | 2.617994 | 1/2 | -sqrt(3)/2 | -1/sqrt(3) |
| 180 | pi | 3.141593 | 0 | -1 | 0 |
| 210 | 7pi/6 | 3.665191 | -1/2 | -sqrt(3)/2 | 1/sqrt(3) |
| 225 | 5pi/4 | 3.926991 | -sqrt(2)/2 | -sqrt(2)/2 | 1 |
| 240 | 4pi/3 | 4.188790 | -sqrt(3)/2 | -1/2 | sqrt(3) |
| 270 | 3pi/2 | 4.712389 | -1 | 0 | undefined |
| 300 | 5pi/3 | 5.235988 | -sqrt(3)/2 | 1/2 | -sqrt(3) |
| 315 | 7pi/4 | 5.497787 | -sqrt(2)/2 | sqrt(2)/2 | -1 |
| 330 | 11pi/6 | 5.759587 | -1/2 | sqrt(3)/2 | -1/sqrt(3) |
| 360 | 2pi | 6.283185 | 0 | 1 | 0 |
Standard angles used in trigonometry, from 0 to 360 degrees, with exact pi-fractions and decimal radians.
Frequently asked questions
What is the formula to convert degrees to radians?
Multiply the angle in degrees by pi/180 (approximately 0.017453). For example, 180 degrees x (pi/180) = pi radians = 3.14159 rad. To go from radians to degrees, multiply by 180/pi (approximately 57.2958).
How many radians are in a full circle?
A full circle is 2*pi radians, which is approximately 6.28318 radians. This equals 360 degrees. A half circle is pi radians (180 degrees), a quarter circle is pi/2 radians (90 degrees), and an eighth of a circle is pi/4 radians (45 degrees).
What are the special angles in radians?
The special angles and their radian equivalents are: 0 degrees = 0, 30 degrees = pi/6, 45 degrees = pi/4, 60 degrees = pi/3, 90 degrees = pi/2, 180 degrees = pi, 270 degrees = 3*pi/2, and 360 degrees = 2*pi. These produce exact values for sin and cos (0, 1, 1/2, sqrt(2)/2, sqrt(3)/2) and are the foundation of the unit circle.
Why do programming languages use radians instead of degrees?
Functions like Math.sin(), Math.cos(), and Math.tan() in JavaScript, Python, C, and most other languages accept angles in radians because the mathematical definition of sine and cosine is based on arc length ratios, which are naturally expressed in radians. Using radians also avoids a scaling factor in derivatives and Taylor series expansions, keeping formulas cleaner. Always convert degrees to radians before passing to these functions.
How do I convert a negative angle or an angle over 360 degrees?
The formula works exactly the same: multiply by pi/180 regardless of the sign or size of the angle. A negative angle like -90 degrees becomes -pi/2 radians (-1.5708 rad), which represents a clockwise quarter turn. An angle like 720 degrees becomes 4*pi radians, representing two full turns. The unit circle behavior repeats every 2*pi radians, so 720 degrees and 360 degrees have the same sine and cosine values.