Angle Converter (degrees, radians, gradians, ...)

How the conversion works

Every unit on this page is defined by how many radians it represents. The input is converted to radians, then from radians into the target unit:

result = value × (radians per source unit) ÷ (radians per target unit)

The key relationships are 2π radians = 360° = 400 gon = 1 full turn.

Why radians are the mathematical default

A degree is an arbitrary choice. The 360-degree circle comes from Babylonian astronomy, where 360 was convenient because it divides evenly by so many numbers and roughly matches the days in a year. There is nothing about circles themselves that produces 360.

A radian is defined geometrically: it is the angle at which the arc length equals the radius. Since a full circle has circumference 2πr, a full turn is exactly 2π radians. No arbitrary constant is involved.

This matters in practice because calculus formulas only work cleanly in radians. The derivative of sin(x) is cos(x) only when x is in radians, in degrees an extra factor of π/180 appears. That is why essentially every programming language's trigonometric functions expect radians:

Math.sin(90) in JavaScript does not give 1, because it interprets 90 as 90 radians. You need Math.sin(90 * Math.PI / 180), or convert first.

Quick reference table

Arcminutes, arcseconds, and GPS coordinates

Degrees subdivide the same way as hours: one degree is 60 arcminutes (written ′), one arcminute is 60 arcseconds (written ″). This sexagesimal system also comes from Babylonian mathematics.

GPS coordinates appear in three formats, which is a frequent source of confusion when copying coordinates between systems:

• Decimal degrees (DD): 48.8584° N
• Degrees and decimal minutes (DDM): 48° 51.504′ N
• Degrees, minutes, seconds (DMS): 48° 51′ 30.24″ N

To convert DMS to decimal degrees, divide the minutes by 60 and the seconds by 3600, then add both to the degrees: 48 + 51/60 + 30.24/3600 = 48.8584°. Getting this wrong typically places a location a few kilometers from where it should be, since one arcminute of latitude is one nautical mile (1.852 km).

Arcseconds also matter in astronomy, where they describe apparent object sizes and telescope resolution. The Moon spans about 1800 arcseconds, and good ground-based seeing is around 1 arcsecond.

Practical examples

Programming and graphics

Rotation in most graphics APIs and game engines uses radians internally, even when the editor UI displays degrees. Convert with radians = degrees × Math.PI / 180. CSS is an exception, it accepts deg, rad, grad, and turn directly, so transform: rotate(0.25turn) is valid and equals 90 degrees.

Roof pitch and slope

Construction expresses slope as a percentage or a ratio rather than an angle. A 100% slope is 45 degrees, not 90, because percentage slope is rise divided by run. To convert, use angle = arctan(slope ÷ 100). A 30% slope is 16.7 degrees.

Surveying with gradians

Gradians, also called gon, divide a right angle into 100 units, which makes 400 gon a full circle. This decimal structure simplifies certain surveying calculations and is still used in parts of continental Europe, particularly in cadastral surveying in France and Germany. Most scientific calculators offer a GRAD mode for this reason.

Compass bearings

Navigation uses degrees measured clockwise from north: 0° is north, 90° east, 180° south, 270° west. Note that this differs from the mathematical convention, where angles are measured counterclockwise from the positive x-axis. Converting between the two requires both a rotation and a direction flip.

Frequently asked questions

How many radians is 360 degrees?

2π radians, approximately 6.2832. Half a circle, 180 degrees, is π radians or about 3.1416.

How do I convert degrees to radians?

Multiply by π/180, which is approximately 0.01745. For 45 degrees: 45 × π/180 = π/4 ≈ 0.7854 radians.

Why do programming languages use radians?

Because calculus and the underlying mathematical definitions of the trigonometric functions assume radians. Using degrees would require an extra conversion factor in every derivative and series expansion.

What is a gradian used for?

Gradians (gon) split a right angle into 100 units, which suits decimal surveying calculations. They are still used in some European surveying contexts but are rare elsewhere.

How do I convert GPS coordinates from DMS to decimal?

Divide minutes by 60, seconds by 3600, and add both to the degrees. For 48° 51′ 30″: 48 + 0.85 + 0.00833 = 48.8583°.

How many arcseconds are in a degree?

3600. One degree is 60 arcminutes, and each arcminute is 60 arcseconds.

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• Length converter, for the arc lengths angles describe
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