How to use the Angle Converter (Degrees, Radians)
- Enter the angle.
- Choose the unit you have.
- Choose the unit you want.
- Read the result and the table of common angles.
Formula used
Result = Value × (degrees per source unit ÷ degrees per target unit). 1 rad = 180 ÷ π degrees ≈ 57.2958°, 1 grad = 0.9°, 1 turn = 360°.
Worked example
Here is the Angle Converter (Degrees, Radians) worked through with the example values shown above, so you can check each step by hand.
| Inputs | |
|---|---|
| Value | 180 |
| From | Degrees (°) |
| To | Radians (rad) |
| Result | |
| Converted value | 3.1415927 |
| Conversion rate | 1 deg = 0.017453293 rad |
Quick reference table
Common values converted with the units selected above.
| From | To |
|---|---|
| 30 deg | 0.52359878 rad |
| 45 deg | 0.78539816 rad |
| 60 deg | 1.0471976 rad |
| 90 deg | 1.5707963 rad |
| 180 deg | 3.1415927 rad |
| 360 deg | 6.2831853 rad |
Degrees and radians
A full circle is 360 degrees, which is 2π radians (6.283185). So 180° is π radians (3.141593), 90° is 1.570796 rad, 60° is 1.047198 rad and 45° is 0.785398 rad.
One radian is the angle where the arc length equals the radius, which is about 57.29578 degrees. To convert degrees to radians multiply by π ÷ 180, and to go back multiply by 180 ÷ π.
Why radians matter
Radians make formulas simpler. The arc length of a sector is radius × angle in radians, and the derivatives of sine and cosine are clean only when the angle is in radians. That is why calculators and programming functions like sin and cos expect radians.
If your calculator gives an unexpected answer for sin(30), check whether it is in degree or radian mode. In radians, sin(30) is about −0.988, not 0.5.
Gradians, turns and small angles
A gradian divides a right angle into 100 parts, so 100 grad is 90° and a full circle is 400 grad. It appears in some surveying instruments. A turn is one full revolution, so half a turn is 180°.
Arcminutes and arcseconds divide a degree into 60 and 3,600 parts. They are used in astronomy, navigation and map coordinates: 1° is 60 arcminutes or 3,600 arcseconds.
Degrees, minutes and seconds
Map coordinates are often written like 40° 26′ 46″. To convert to decimal degrees add the degrees, the minutes divided by 60, and the seconds divided by 3,600. For this example 40 + 0.4333 + 0.0128 = 40.4461°.
Use the converter for each part, or do it by hand with those divisors.
Common angles at a glance
The angles you meet most often are 30° (π/6 rad), 45° (π/4), 60° (π/3), 90° (π/2), 180° (π) and 360° (2π). Memorising these makes trigonometry far quicker, because the sine, cosine and tangent of these angles have exact values.
A compass bearing is an angle measured clockwise from north in degrees, from 0 to 360. East is 90°, south 180° and west 270°. Clock hands also make angles: the minute hand moves 6° a minute and the hour hand 0.5° a minute, which is a classic puzzle source.
Frequently asked questions
How do I convert degrees to radians?
Multiply by π ÷ 180. For example, 180° is 3.141593 radians.
How many degrees is one radian?
About 57.29578 degrees.
How many radians are in a full circle?
2π, or 6.283185 radians.
What is a gradian?
A unit where a right angle is 100 grad, so a full circle is 400 grad. It is also called a gon.
Why does my calculator give a strange sine value?
It may be in the wrong mode. Check whether it uses degrees or radians for your problem.
How many arcseconds are in a degree?
3,600. A degree has 60 arcminutes, and each arcminute has 60 arcseconds.
This calculator gives estimates for information only. Actual figures from a bank, employer, tax authority or doctor may differ. It is not financial, tax, legal or medical advice.