Multiply degrees by π ÷ 180 to get radians; a full turn is 360 degrees, 2π radians, 400 gradians or 1 turn. Enter an angle in any of six units to see all of them at once, with degrees, minutes and seconds.
The formula
One full turn
1 turn = 360° = 2π rad = 400 grad = 21,600′ = 1,296,000″Degrees to radians
radians = degrees × π ÷ 180Radians to degrees
degrees = radians × 180 ÷ πDegrees to gradians
gradians = degrees × 10 ÷ 9Degrees, minutes, seconds
1° = 60′ = 3,600″Every conversion goes through turns, using whole numbers of each unit in a turn, so only the radian involves π.
Worked example
And for 1 radian: 1 ÷ 2π = 0.1591549 turn, which is 57.29578°, or 57° 17′ 44.81″.
When to use it, and the mistakes to avoid
Use it for trigonometry and calculus (radians), surveying and navigation (degrees, minutes, seconds and gradians), geometry homework, and setting a calculator or a program to the unit it expects.
- Using the wrong mode. A calculator in degrees mode and one in radians mode give different answers for the same sine: check which you are in.
- Mixing up minutes of angle and minutes of time. An arcminute is 1/60 of a degree, not of an hour.
- Rounding π too early. 3.14 gives 1.5700 rad for 90° where π gives 1.5708. Keep full precision until the end.
- Treating 450° as 90° in a rotation count. They end in the same place, but one is a turn and a quarter.
- Reading a negative angle as an error. It is a rotation the other way.
FAQ
How do I convert degrees to radians?
Multiply by π ÷ 180, which is about 0.017453. So 90° is 90 × π ÷ 180 = π ÷ 2 = 1.570796 radians, and 180° is π radians. To go the other way, multiply radians by 180 ÷ π: 1 radian is 57.29578°.
What is a radian?
The angle at the centre of a circle that cuts off an arc as long as the circle's radius. A full circle is 2π radians, because its circumference is 2π times the radius. Radians are the natural unit in maths and physics because the formulas for arc length (r × θ) and for the derivatives of sine and cosine are simplest in them.
What is a gradian?
A unit that divides a right angle into 100, so a full circle is 400 gradians (also called gons or grades). It is used in some surveying and civil engineering, because a right angle is a round 100 and slopes convert easily to percentages. 90° is exactly 100 gradians.
How do I read degrees, minutes and seconds?
A degree is split into 60 arcminutes (′) and each arcminute into 60 arcseconds (″), so one arcsecond is 1/3,600 of a degree. 45° 30′ 0″ is 45.5°. Maps, navigation and astronomy use this form: a latitude of 12° 58′ 12″ is 12.97°. The calculator shows any angle this way.
Why does an angle over 360° or below 0° still work?
Because an angle is a rotation, and a rotation can go round more than once or the other way. 450° is a full turn plus 90°, and −90° is a quarter turn the other way, which ends where 270° does. The calculator converts the angle you typed as it is, and the scale shows where it ends up on one circle.