Degrees To Radians Converter
180 degrees is pi radians, about 3.1416, because a half turn is pi radians.
Quick answer: Multiply the degrees by pi and divide by 180, which is about 0.017453. For example, 180 deg is 3.142 rad, 360 deg is 6.283 rad and 90 deg is 1.571 rad.
The degrees to radians formula
Radians equal degrees times pi divided by 180.
The factor pi divided by 180 is about 0.0174533.
Take 180 deg. 180 × 0.01745329 gives 3.142 rad.
Take 360 deg. 360 × 0.01745329 gives 6.283 rad.
Why pi over 180
A half turn is 180 degrees and also pi radians (Encyclopedia of Mathematics: Radian). So one degree is pi divided by 180 radians.
Multiply the degrees by that number to get radians. A full turn of 360 degrees is 2 pi radians (Wikipedia: Turn (angle)).
How to convert degrees to radians step by step
- Read the angle in degrees.
- Multiply it by 0.0174533.
- Write the answer in radians.
- To keep it exact, leave the answer as a fraction of pi.
Example 1: 180 deg
- Start with 180 deg.
- 180 × 0.01745329 = 3.142.
- The angle is 3.142 rad.
180 deg = 3.142 rad
Example 2: 360 deg
- Start with 360 deg.
- 360 × 0.01745329 = 6.283.
- The angle is 6.283 rad.
360 deg = 6.283 rad
Example 3: 90 deg
- Start with 90 deg.
- 90 × 0.01745329 = 1.571.
- The angle is 1.571 rad.
90 deg = 1.571 rad
Example 4: 900 deg
- Start with 900 deg.
- 900 × 0.01745329 = 15.708.
- The angle is 15.708 rad.
900 deg = 15.708 rad
Example 5: 1,800 deg
- Start with 1,800 deg.
- 1,800 × 0.01745329 = 31.416.
- The angle is 31.416 rad.
1,800 deg = 31.416 rad
Example 6: 18 deg
- Start with 18 deg.
- 18 × 0.01745329 = 0.3142.
- The angle is 0.3142 rad.
18 deg = 0.3142 rad
Round only at the end. Rounding in the middle of a sum adds small errors that grow.
degrees to radians conversion table
| Degrees (deg) | Radians (rad) | Turns |
|---|---|---|
| 1.8 deg | 0.0314 rad | 0.005 Turns |
| 18 deg | 0.3142 rad | 0.05 Turns |
| 90 deg | 1.571 rad | 0.25 Turns |
| 180 deg | 3.142 rad | 0.5 Turns |
| 360 deg | 6.283 rad | 1 Turns |
| 900 deg | 15.708 rad | 2.5 Turns |
| 1,800 deg | 31.416 rad | 5 Turns |
| 4,500 deg | 78.54 rad | 12.5 Turns |
| 18,000 deg | 314.16 rad | 50 Turns |
The highlighted row is the default of the calculator. The last column gives Turns.
A quick mental shortcut for degrees to radians
Think of 1 degree as about 0.0175 radians. So 10 degrees is about 0.175 radians.
Or write the angle as a fraction of 180. 90 degrees is half of pi.
Radians as multiples of pi
Many answers are cleaner as a multiple of pi. 30 degrees is pi over 6 and 45 degrees is pi over 4.
60 degrees is pi over 3, and 90 degrees is pi over 2.
Where radians are used
Maths, physics and computer code want angles in radians. Sine, cosine and angular speed all use them.
A plan or a map gives degrees, so you convert at the start.
Quick conversion answers
| Conversion | Result |
|---|---|
| 18 deg | 0.3142 rad |
| 180 deg | 3.142 rad |
| 1,800 deg | 31.416 rad |
| 18,000 deg | 314.16 rad |
Where this conversion is used
Students and coders turn a degree angle into radians for a formula.
Engineers convert a rotation for software.
Common mistakes to avoid
- Dividing by 0.0174533. That goes from radians to degrees.
- Using 180 over pi. That is the other direction.
- Rounding pi too early. Round only at the end.
- Calculator in the wrong mode. Check radian or degree mode.
Related converters
More values at a glance
3.6 deg is 0.06283 rad. 36 deg is 0.6283 rad. 54 deg is 0.9425 rad.
135 deg is 2.356 rad. 270 deg is 4.712 rad. 540 deg is 9.425 rad.
720 deg is 12.57 rad. 3,600 deg is 62.83 rad. 9,000 deg is 157.1 rad.
More worked values
1 deg is 0.017453 rad. 2 deg is 0.034907 rad. 4 deg is 0.069813 rad.
8 deg is 0.139626 rad. 15 deg is 0.261799 rad. 25 deg is 0.436332 rad.
40 deg is 0.698132 rad. 75 deg is 1.308997 rad. 100 deg is 1.745329 rad.
Going the other way
To turn rad back into deg, do the opposite step.
1 rad is 57.29578 deg. 5 rad is 286.478898 deg. 10 rad is 572.957795 deg.
20 rad is 1,145.91559 deg. 50 rad is 2,864.788976 deg. 100 rad is 5,729.577951 deg.
250 rad is 14,323.944878 deg. 500 rad is 28,647.889757 deg.
A full worked example
Take 7.5 deg. Multiply by 0.017453 to get 0.1309 rad.
Now check it. Turn 0.1309 rad back into deg and you should land on 7.5 again.
If you do not, look for a slipped decimal point or a swapped unit.
Rounding the answer
Most work uses two to four decimals. For example, 1.2345 deg is 0.021546 rad, which rounds to 0.02 rad.
Round only at the end. Rounding early can add a small error to the final number.
Small and large values
0.1 deg is 0.001745 rad. 0.25 deg is 0.004363 rad. 0.5 deg is 0.008727 rad.
3 deg is 0.05236 rad. 6 deg is 0.10472 rad. 12 deg is 0.20944 rad.
30 deg is 0.523599 rad. 60 deg is 1.047198 rad. 150 deg is 2.617994 rad.
200 deg is 3.490659 rad. 300 deg is 5.235988 rad. 400 deg is 6.981317 rad.
More values in the reverse direction
2 rad is 114.591559 deg. 3 rad is 171.887339 deg. 25 rad is 1,432.394488 deg.
30 rad is 1,718.873385 deg. 40 rad is 2,291.831181 deg. 60 rad is 3,437.746771 deg.
75 rad is 4,297.183463 deg. 200 rad is 11,459.155903 deg. 400 rad is 22,918.311805 deg.
1,000 rad is 57,295.779513 deg.
Checking the answer
Multiply the radians by 57.29578. You should get the degrees you started with.
For 3.1416 radians, that is about 180 degrees.
Frequently asked questions
How many Radians (rad) is 1 deg?
1 deg is 0.01745329 rad.
How many Radians (rad) is 360 deg?
360 deg is 6.28319 rad.
How many Degrees (deg) is 1 rad?
1 rad is 57.29578 deg.
How many radians is 1 degree?
About 0.017453 radians.
How many radians is 180 degrees?
Pi, about 3.1416.
How do I go from radians to degrees?
Multiply by 180 and divide by pi.
Sources
- Encyclopedia of Mathematics: Radian (encyclopediaofmath.org)
- Wikipedia: Turn (angle) (en.wikipedia.org)
