Angle Converter — Degrees, Radians, DMS & Turns

Background.

Angles are ratios, but the units used to write them come from several different traditions. The radian is the coherent SI unit: one radian is the angle that cuts an arc equal in length to the circle's radius. Degrees divide a full turn into 360 parts. Gradians, also called gon, divide it into 400. A turn or revolution names the whole circle directly, while “π radians” is a convenient way to preserve familiar fractions such as π/2 without hiding the factor of π in a decimal.

BIPM gives the relationship that connects the entire page: 360 degrees equals 2π radians exactly. From that identity, a half turn is 180 degrees or π radians, a right angle is 90 degrees or π/2 radians, and one degree is π/180 radians. NIST's conversion table independently gives one gon as 0.9 degree and one revolution as 2π radians. These are definitions, not measured constants, so no uncertainty or locale-specific factor is involved.

Degrees-minutes-seconds, or DMS, is another notation for degrees rather than a different kind of angle. It is sexagesimal: one degree contains 60 arcminutes and one arcminute contains 60 arcseconds. Decimal 12.5 degrees is therefore 12 degrees 30 minutes 0 seconds. The page always shows DMS beside the numeric unit results and correctly carries a seconds value that rounds to 60 into the next minute. A negative sign applies to the complete angle; minutes and seconds remain non-negative components.

Choose the unit your number already uses. The calculator first converts it to degrees, then derives radians, gradians, turns and π-radian multiples from that common value. Decimal.js performs every arithmetic step, including π, and numeric outputs cross the display boundary after rounding to ten decimal places. The DMS string is assembled from the unrounded degree value so it does not inherit an earlier rounded intermediate.

Angles are not automatically bearings. This page deliberately preserves −30 degrees, 390 degrees and 2.5 turns as entered rather than wrapping them into a 0–360 range. In geometry, rotation, phase and animation, the number of completed turns or direction of rotation may matter. A navigation bearing has additional normalization rules, and a latitude or longitude has limits plus a cardinal hemisphere. Use the coordinate converter when N/S/E/W is part of the meaning.

Likewise, DMS precision should not be mistaken for physical accuracy. Writing many arcsecond decimals only states the numerical angle more finely. In geographic coordinates, the ground distance represented by one longitude second changes with latitude, and the accuracy can never exceed the source measurement. The converter preserves and rearranges the input; it does not improve a survey, sensor or map.

The π-radian mode accepts the coefficient rather than the product. Enter 0.5 to mean π/2 radians, not the decimal number 0.5 radians. This mode is useful for textbook and trigonometric values, while the ordinary radians mode is appropriate for a measured or already-decimal radian value. Both produce the same outputs when their inputs describe the same angle.

This separation of value, unit and context is the key to reliable angle conversion. The formulas are exact, the output set makes cross-checking easy, and the page refuses to invent coordinate semantics that an angle alone does not carry.

What is angle converter?

A plane-angle converter preserves an angle while changing the unit used to express it. The degree and gradian partition one turn into 360 and 400 equal parts. The radian relates arc length to radius, making one turn 2π radians. DMS subdivides one degree into 60 arcminutes and each arcminute into 60 arcseconds. Because these relationships are definitional, conversion requires only multiplication or division by exact factors and π.

How to use this calculator.

1. 01Enter the angle and choose the unit it currently uses.
2. 02For a value written as a multiple of π, enter only its coefficient and select the π-radians mode.
3. 03Read decimal degrees as the primary result and use radians, gradians, turns or DMS from the breakdown.
4. 04Keep the sign on the whole DMS result; do not make minutes or seconds negative separately.
5. 05Use a coordinate-specific tool if the number represents latitude, longitude or a bearing that must be range-checked or wrapped.

The formula.

Every mode first becomes degrees. Ordinary radians multiply by 180/π. Gradians multiply by the exact NIST factor 0.9 degree per gon. Turns multiply by 360. A coefficient of π radians multiplies by 180, because π radians equals 180 degrees. Once degrees are known, the reverse factors produce every result.

DMS decomposition uses the absolute value. The integer part is degrees. Multiply the remaining fraction by 60; its integer part is minutes. Multiply the remaining minute fraction by 60 for seconds. The original sign is added once before the degree component. Seconds are rounded to ten decimal places; if that result is 60, it becomes zero and one minute is carried, with the same carry from 60 minutes into one degree.

For example, 12.345678 degrees leaves 0.345678 after extracting 12. Multiplying by 60 gives 20.74068, so minutes are 20. The remaining 0.74068×60 is 44.4408 seconds. The DMS result is 12° 20′ 44.4408″. The radian result follows independently as 12.345678×π/180.

A worked example.

Frequently asked questions.

References& sources.

1. [1]BIPM SI Brochure, 9th edition updated 2026, §5.4.8 defines the radian and states 360° = 2π rad.
2. [2]NIST SP 811 Appendix B.9 publishes degree, gon, revolution, arcminute and arcsecond angle conversion factors.
3. [3]NIST SP 330 §5 confirms the coherent radian unit and the symbols for degrees, arcminutes and arcseconds.
4. [4]USGS and NASA Technical Methods 11-E1 §§2.33–2.35 gives independent decimal-degree and DMS conversion procedures.
5. [5]NGA's coordinate-system reference explains the sexagesimal division into 60 minutes per degree and 60 seconds per minute.