When to use this converter
Use it for trigonometry homework, programming (most math libraries expect radians), CAD and engineering drawings, or surveying instruments that read in gradians. For trigonometry itself, the online scientific calculator works in degrees or radians.
The units
A full turn is 360 degrees, 2π radians or 400 gradians. The radian is the SI unit: it is the angle at which the arc length equals the radius. Gradians, also called gons, divide a right angle into 100 parts and are used mainly in surveying. Arcminutes (′) and arcseconds (″) split a degree into 60 and 3,600 parts, the way map coordinates and telescope specs are written.
Negative and large angles
Angles have a direction, so negative values are allowed and converted as-is: −90° is −π/2 rad. Values beyond a full turn are also converted without wrapping, so 720° stays 4π rad.
Worked examples
Programming: to rotate a sprite 30°, pass 30 × π/180 ≈ 0.5236 rad to the math library. To show the result of atan2 to a user, multiply the radians by 180/π. Spreadsheets work the same way — Excel’s SIN() expects radians, so wrap degrees in RADIANS(). The degrees to radians and radians to degrees pages list common angles.
Maps and astronomy: a coordinate written 40° 26′ 46″ is 40 + 26/60 + 46/3,600 ≈ 40.4461°. A minute of latitude is about one nautical mile, which is why the nautical mile and the knot exist.
Surveying: 100 gon is a right angle (90°) and 1 gon is 0.9°. Slope in percent is a different measure: a 100% grade is a 45° angle, not 90°.
Navigation: compass bearings run clockwise from north, 0° to 360°, while math angles run counterclockwise from the positive x-axis. Convert the reference direction first, then the unit, when moving between the two conventions.
A protractor reads whole degrees; for finer work, record arcminutes or decimal degrees and convert only once at the end.