Angle Converter

Last updated: 2026-05-14

What is an angle unit?

An angle measures the rotation between two rays sharing a common endpoint. Different fields use different units to express that rotation, and being able to convert between them is essential in mathematics, engineering, navigation, and programming.

The four units

Degrees (°) are the oldest and most familiar unit. A full circle is 360°, a right angle is 90°, and a straight angle is 180°. The 360-degree convention comes from ancient Babylonian astronomy, where the number 360 was chosen because it is divisible by many integers (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180).

Radians (rad) are the SI unit for angles and the natural unit for calculus. One radian is the angle subtended at the center of a circle by an arc whose length equals the radius. A full circle is exactly 2π radians (≈ 6.28318). Because the radian is dimensionless (arc length divided by radius), trigonometric derivatives come out clean without extra factors — d/dx [sin(x)] = cos(x) holds only when x is in radians.

Gradians (grad), also called gons or grades, divide a full circle into 400 equal parts. A right angle is exactly 100 gradians. Proposed during the French Revolution as part of the metric system, gradians are still used in surveying, civil engineering, and some military contexts because they simplify calculations involving right angles.

Turns (also called revolutions or cycles) express angle as a fraction of a full rotation. One turn equals 360°, 2π radians, or 400 gradians. Turns are increasingly popular in programming and API design because they make the "full circle" concept explicit — a rotation of 0.5 turns is always a half-circle, regardless of any convention.

Conversion formulas

From	To degrees	To radians	To gradians	To turns
1 degree	1°	π/180	10/9	1/360
1 radian	180/π	1	200/π	1/(2π)
1 gradian	9/10	π/200	1	1/400
1 turn	360	2π	400	1

How to use this calculator

Enter any angle value in any field. All other units update automatically. For example:

• Type 90 in the Degrees field → Radians shows 1.5708, Gradians shows 100, Turns shows 0.25.
• Type 1 in the Turns field → all other fields update to reflect one full revolution.

Frequently Asked Questions (FAQ)

What is the difference between degrees and radians?

Degrees divide a full circle into 360 equal parts — a historical convention from Babylonian astronomy. Radians measure the angle by the arc length it subtends on a unit circle: a full circle is 2π radians (≈ 6.2832). Radians are the natural unit for calculus and physics because they avoid scale factors in derivatives of trigonometric functions.

What is a gradian (grad)?

A gradian (also called gon or grade) divides a full circle into 400 equal parts, placing 100 gradians at a right angle. Introduced during the French Revolution alongside the metric system, gradians are still used in surveying and civil engineering in many European countries because they make right-angle arithmetic simpler.

What are the full-circle equivalents in each unit?

A full 360° rotation equals exactly 2π radians (≈ 6.2832), 400 gradians, and 1 turn. A right angle is 90°, π/2 rad, 100 grad, or 0.25 turn. These equivalences make mental-math conversions between degrees and gradians particularly easy — just multiply or divide by 10/9.

Why do programmers often prefer turns or radians over degrees?

In code, angles are almost always passed to math functions (sin, cos, atan2) that expect radians — so working in radians avoids constant × π/180 conversions. Some modern APIs (CSS conic-gradient, game engines) accept turns (e.g. 0.5turn) because the intent is immediately clear. Degrees remain best for human-facing displays and navigation where people expect familiar numbers.

Are there other angle units not listed here?

Yes. Arcminutes (1° = 60 arcminutes) and arcseconds (1° = 3600 arcseconds) are used in astronomy and GPS coordinates. Milliradians (mrad) are used in ballistics and optics. Hours of arc (1 hour = 15°) appear in celestial coordinates. This converter focuses on the four most commonly needed general-purpose units.