A radian is the angle subtended at the centre of a circle by an arc equal in length to the radius. One full circle is 2π radians (≈ 6.2832 rad). 1 radian ≈ 57.2958 degrees.

How many degrees in a circle?

A full circle contains 360 degrees, 2π radians, 400 gradians, or 1 turn. This division into 360 originated with ancient Babylonian mathematics.

A gradian (also called gon or grade) divides a right angle into 100 parts, making a full circle 400 gradians. It is used in surveying and some European countries. 1 gradian = 0.9 degrees.

When do you use radians vs degrees?

Radians are the standard in mathematics, physics, and programming because they simplify calculus formulas (derivatives of trig functions). Degrees are used in everyday measurement, navigation, and engineering drawings.

Angle Converter

Convert between Degrees, Radians, Gradians, Turns, Arcminutes, and Arcseconds instantly.

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How to Convert Angle Units

Angle conversion works by converting any input to the base unit (degrees), then to the target unit. Angles are fundamental in geometry, trigonometry, navigation, astronomy, and engineering.

Common Conversion Factors

From	To	Multiply by
Degree	Radian	π/180 ≈ 0.01745
Radian	Degree	180/π ≈ 57.2958
Degree	Gradian	10/9 — 1.1111
Turn	Degree	360
Degree	Arcminute	60
Degree	Arcsecond	3,600
Turn	Radian	2p — 6.2832

Angles in Everyday Life

Navigation: Compass bearings use degrees (0—360°). Mathematics: Calculus and physics prefer radians because derivative formulas (sin, cos) are simplest in radians. Surveying: Some countries use gradians, where a right angle is exactly 100 grad. Astronomy: Arcminutes and arcseconds measure tiny angular separations between stars.

Common Angles Reference

Degrees	Radians	Gradians	Turns
0°	0	0	0
30°	π/6 ≈ 0.5236	33.33	0.0833
45°	π/4 ≈ 0.7854	50	0.125
90°	p/2 — 1.5708	100	0.25
180°	π ≈ 3.1416	200	0.5
360°	2p — 6.2832	400	1

Frequently Asked Questions

A radian is the angle formed when an arc length equals the circle's radius. One full revolution is 2π radians (≈ 6.2832). Since 1 radian ≈ 57.296°, radians are larger than degrees. They are the natural unit in calculus and physics.

A full circle contains exactly 360 degrees. This convention originated with the ancient Babylonians, who used a base-60 number system. A circle also equals 2π radians, 400 gradians, or 1 turn.

A gradian (gon or grade) divides a right angle into 100 equal parts, making a full circle 400 gradians. This decimal-friendly system is used in surveying and land measurement, particularly in France and some other European countries.

Use radians in mathematics, physics, and programming — calculus formulas (derivatives, Taylor series) only work cleanly in radians. Use degrees for everyday measurement, navigation, and architecture. Most programming languages' trig functions (sin, cos, tan) expect radians.

Three ways to talk about how far around something is

Degrees, radians, and gradians are three competing conventions for measuring angular distance. Each one carves a full circle into a different number of equal parts: 360 for degrees, 2π for radians, 400 for gradians. Convert between them with three constants: 180° = π rad; 1 rad = 180 / π ≈ 57.2958°; 1 grad = 0.9°.

Where each is used

Degrees dominate everyday and trade contexts: navigation, surveying, construction, hand tools, geometry classes.
Radians are the natural unit in calculus and physics because the derivative formulas of sin and cos work cleanly only in radians.
Gradians persist in some surveying and military contexts, especially in continental Europe, because a right angle is exactly 100 grad.

Common conversions worth remembering

30° = π/6 rad. 45° = π/4 rad. 60° = π/3 rad. 90° = π/2 rad. 180° = π rad. 360° = 2π rad. For programming, almost every standard library uses radians (Math.sin in JavaScript, math.sin in Python). If you are passing degree values from a UI, multiply by π/180 first.

Bearings, headings, and quadrants

Navigation bearings are usually given in degrees clockwise from north (0–360), but some surveying uses the quadrant convention (N 30° E means 30 degrees east of north). Mixing them up is one of the most common errors in route planning.