Angle Converter — Degrees, Radians, DMS & Turns

Background.

Type sin 30 into a scientific calculator left in radian mode and it answers −0.988 instead of 0.5. The arithmetic is fine: the calculator read 30 as 30 radians, which is more than four and a half full turns. Most angle mistakes are of this kind. The number survives intact while its unit quietly changes underneath it.

Five units are in common use for the same quantity. Degrees divide a full turn into 360 parts and dominate everyday geometry, navigation and construction. Radians measure an angle by the arc it cuts on a circle, counted in radii, and are the unit of calculus, physics and the sin and cos functions of every programming language. Gradians, or gon, split the turn into 400 so that a right angle is a round 100. Turns count whole revolutions. Multiples of π keep textbook angles such as π/6 recognisable instead of burying them in decimals. Degrees-minutes-seconds is not a sixth unit but a way of writing degrees in base 60, still standard for latitude and longitude, astronomy and surveying.

One identity links them all: 360° = 2π rad = 400 gon = 1 turn. It is a definition rather than a measurement, so every conversion is exact apart from the digits of π. Enter an angle in any of the five units and all five come back, along with the degrees-minutes-seconds form.

The angle is converted as entered and never wrapped. −30° stays −30°, and 450° stays 450°, or 1.25 turns, instead of becoming 90°, because in rotation, phase and animation the count of completed turns and the direction both carry meaning. A compass bearing or a map coordinate adds rules of its own, a permitted range and a hemisphere, that an angle by itself does not have.

What is angle converter?

A plane angle measures how far one ray is turned from another, and every unit for it is a way of slicing a full turn. The radian is the SI unit: an angle of one radian cuts an arc exactly as long as the circle's radius, so a full circle, whose circumference is 2π radii, is 2π rad, and one radian is about 57.2958°. Because it is a ratio of two lengths it is dimensionless, which is why formulas such as arc length s = rθ and speed v = ωr hold only when θ and ω are in radians. On a wheel of radius 10 cm, a 30° turn moves a point on the rim 10 × 0.5236 = 5.236 cm, because 30° is 0.5236 rad; putting 30 into the formula instead would give 300 cm. The degree is 1/360 of a turn, the gradian 1/400 and the turn the whole circle. Degrees are subdivided into 60 arcminutes (′) and each arcminute into 60 arcseconds (″), so 12.5° is written 12° 30′ 0″.

How to use this calculator.

1. 01Type the angle into Angle and pick its current unit under Input unit. Negative and multi-turn values are fine.
2. 02For an angle written as a fraction of π, choose Multiples of π radians and enter only the coefficient: 0.5 for π/2, 0.25 for π/4, 2 for a full circle. A plain decimal such as 1.5708 belongs in Radians.
3. 03To start from degrees, minutes and seconds, combine them first as degrees + minutes ÷ 60 + seconds ÷ 3,600 and enter the total in decimal degrees; 40° 26′ 46″ becomes 40.4461°.
4. 04Read whichever unit you need. Decimal degrees leads, followed by radians, gradians, turns, π multiples and the DMS string.
5. 05If the angle is a latitude, longitude or bearing, carry the sign or hemisphere over yourself: a negative DMS result for a longitude means west.

The formula.

The input is first turned into degrees, and every other unit is derived from that one value. Radians are multiplied by 180/π, gradians by 0.9 (a gon is exactly 0.9°), turns by 360 and π multiples by 180, since π rad is 180°. Going out, the factors run in reverse: radians = degrees × π ÷ 180, gradians = degrees ÷ 0.9, turns = degrees ÷ 360 and multiples of π = degrees ÷ 180. All of them follow from the single identity 360° = 2π rad = 400 gon = 1 turn, as set out in the BIPM's SI Brochure and NIST's conversion tables.

Degrees-minutes-seconds is built from the size of the angle, ignoring its sign. The whole-number part gives the degrees. The leftover fraction times 60 gives arcminutes, and its whole-number part is kept. The fraction of a minute that remains, times 60, gives arcseconds. If the seconds would round up to 60 they become 0 and a minute is carried, and 60 minutes carry into the next degree in the same way, so a result never reads 59′ 60″. The minus sign is placed once, in front of the degrees, because it belongs to the whole angle: −12° 20′ 44″ means −(12° + 20′ + 44″), which is −12.3456°, not −12° plus 20′ and 44″, which would be −11.6544°. Software that reads DMS back in has to follow the same convention.

Radian values are almost never tidy. Because π is irrational, every whole number of degrees other than zero becomes an endless decimal in radians: 90° is 1.5708 rad to four places, and only π/2 is exact. The π-multiples output exists for that reason. Dividing degrees by 180 turns familiar angles into short fractions of π, so 30° is 0.1667π (π/6), 45° is 0.25π and 120° is 0.6667π (2π/3), forms that match the way textbooks and trigonometric tables write them.

The reverse of that split is decimal degrees = degrees + minutes ÷ 60 + seconds ÷ 3,600, with the sign applied to the total. For rough mental checks, 1 rad ≈ 57.2958° and 1° ≈ 0.017453 rad, so radians are close to degrees divided by 57.3.

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.