Angle units: overview and conversion factors
Degrees, radians, gon and arcminutes – mathematics, navigation and surveying
An angle describes the opening between two rays that share a common starting point. The most common unit is the degree (°), where a full circle is divided into 360 degrees. This division comes from ancient Babylon, which used a sexagesimal (base-60) number system. Because 360 is divisible by many numbers (1, 2, 3, 4, 5, 6, 8, 9, 10, 12...), it produces a practical subdivision. Right angles (90°), obtuse angles (90°–180°) and straight angles (180°) are intuitive everyday concepts.
In mathematics and physics, the radian (rad) is the dominant "natural" angle unit. One radian is the angle at which the arc length of a circle equals its radius. A full circle has 2π ≈ 6.2832 radians. The radian is a dimensionless unit (arc length divided by radius). Trigonometric functions (sin, cos, tan) are given in radians by default in calculus and signal processing, because the derivatives are then especially simple: d/dx(sin x) = cos x.
The gon (gon), also called grad or gradian, divides the full circle into 400 equal parts. That means a right angle equals exactly 100 gon – an advantage for decimal calculations. Gon is used mainly in surveying, cartography and construction planning. Theodolites – the angle-measuring instruments used by surveyors – often display their readings in gon. Gon is especially common in Switzerland and France.
The arcminute (′) is 1/60 of a degree. An arcminute measured along the Earth's meridian corresponds to one nautical mile (1,852 m) – hence the name. 1° = 60 arcminutes = 3,600 arcseconds. GPS coordinates are often given in degrees, minutes and seconds (DMS): e.g. 48° 8′ 14″ N, 11° 34′ 32″ E for Munich. In astronomy, angular distances between celestial objects are expressed in arcminutes and arcseconds. The human eye can resolve angular detail down to about 1 arcminute.
In everyday life and engineering, angles come up in many contexts: roof pitches are given in degrees (typically 15°–45°), road gradients as a percentage (tan(angle) × 100). In navigation, bearings are given in degrees (0°–360°, north = 0°). Saw blades are set to specific degrees, as are drilling angles. Our angle converter lets you quickly convert between all common angle units for mathematics, engineering and navigation.