Angle Converter

How it works

Every angle is first expressed in radians, the SI unit, then divided by the size of the unit you are converting to. A full turn is 2π radians, 360 degrees, 400 gradians, 21,600 arcminutes or 1,296,000 arcseconds, so each unit is a fixed fraction of a turn and the conversion is a single multiplication.

Degrees, minutes and seconds (DMS) split a degree sexagesimally: 1° = 60′ and 1′ = 60″, so 30° 15′ 50.4″ is 30 + 15/60 + 50.4/3600 = 30.264°. Choose "Degrees, minutes and seconds" to enter an angle that way, for example a latitude from a map or a bearing from a survey. Whatever you enter, the DMS line in the results shows the angle in that form, with seconds to two decimal places.

A negative angle in DMS carries its sign on the degrees field: −30° 15′ means −(30 + 15/60). Minutes and seconds are always non-negative.

Formula

radians   = value × radians per source unit
converted = radians ÷ radians per target unit
π rad = 180° = 200 grad = ½ turn;  1 rad = 1000 mrad
1° = 60′ = 3600″
DMS → decimal: ° + ′/60 + ″/3600
decimal → DMS: ° = floor(x), ′ = floor(60 × frac), ″ = 3600 × remainder

Example

45°, the default, is 45 × π/180 = 0.785398 rad. It is also 50 grad, 0.125 turn, 2,700′, 162,000″ and 785.398163 mrad, and reads 45° 0′ 0″ in DMS.

A latitude of 30° 15′ 50.4″ is 30 + 15/60 + 50.4/3600 = 30.264°, or 0.528206 rad. Going the other way, 1 radian is 180/π = 57.295780°, which is 57° 17′ 44.81″.

Assumptions and limitations

• Gradians (also called gons) divide a right angle into 100, so 400 grad = 360°. Milliradians are the SI milliradian (1/1000 rad, 6,283.19 per turn), not the NATO mil (6,400 per turn) or the Soviet mil (6,000 per turn).
• Angles are not reduced modulo a full turn: 450° converts to 2.5π rad, not 0.5π. Reduce the angle yourself if you need the equivalent angle within one turn.
• DMS seconds are rounded to two decimal places (about 0.0000028°), with a carry into the minutes and degrees when the rounding reaches 60″. In DMS entry, degrees must be a whole number; enter 12.5° in decimal mode instead.
• Results are shown to six decimal places; the underlying arithmetic is double precision.

Frequently asked questions