Angle Converter – Degrees, Radians, Gradians & More

Degrees (°) - The Most Familiar Unit

The degree is the most widely recognised unit of angle measurement. A full circle is divided into 360 degrees, a convention that dates back over 2,000 years to Babylonian astronomy. Each degree is further divided into 60 arcminutes, and each arcminute into 60 arcseconds. Degrees are used in everyday life, navigation, geography (latitude and longitude), construction, and most school-level mathematics. The symbol for degree is °.

Key facts: 360° = full circle | 180° = straight angle | 90° = right angle | 1° = π/180 radians ≈ 0.017453 rad

Radians (rad) - The SI Unit for Angles

The radian is the SI-derived unit of angular measure and the preferred unit in mathematics, physics, and engineering. One radian is defined as the angle subtended at the centre of a circle by an arc whose length equals the circle's radius. A full revolution equals exactly 2π radians (approximately 6.28318 rad). Radians simplify many formulas - for instance, the arc length formula is simply s = rθ (where θ is in radians), and the derivatives of sin and cos are cleanest in radians.

Key facts: 2π rad = 360° | π rad = 180° | π/2 rad = 90° | 1 rad ≈ 57.2958°

Degrees to radians formula: radians = degrees × (π ÷ 180)
Radians to degrees formula: degrees = radians × (180 ÷ π)

Gradians / Gons (grad / gon) - Surveyor's Choice

The gradian (also called gon or grade) divides a right angle into exactly 100 equal parts, meaning a full circle contains 400 gradians. This decimal structure makes gradians especially convenient in surveying and civil engineering, where percentage grades and slope calculations align neatly with base-10 arithmetic. The gradian system was developed in France during the metric revolution of the 18th century. The symbols grad and gon are used interchangeably - they are identical units.

Key facts: 400 grad = 360° | 100 grad = 90° | 1 grad = 0.9° | 1° = 1.1111 grad

Degrees to gradians formula: gradians = degrees × (400 ÷ 360) = degrees × (10 ÷ 9)
Gradians to degrees formula: degrees = gradians × 0.9

Turns / Revolutions - Full Rotations

A turn (also called a revolution, cycle, or full rotation) represents one complete 360° circuit. Turns are intuitive for expressing rotational motion - gear ratios, motor speeds (RPM), and clock hand positions. In computer graphics and game development, angles are sometimes specified as fractions of a turn (e.g. 0.25 turns = 90°). One turn equals 2π radians or 400 gradians.

Key facts: 1 turn = 360° = 2π rad = 400 grad | 0.5 turn = 180° | 0.25 turn = 90°

Arcminutes (′) - Precision Navigation & Astronomy

An arcminute (symbol ′, also written "arcmin" or "minute of arc") is 1/60 of one degree. Because one arcminute of latitude at the Earth's equator corresponds to approximately one nautical mile (1,852 m), arcminutes are fundamental to marine and aviation navigation. Astronomers use arcminutes to specify the apparent size of celestial objects - for example, the full Moon subtends about 31 arcminutes (roughly 0.5°). Arcminutes also appear in optics to describe visual acuity: normal human vision resolves detail down to about 1 arcminute.

Key facts: 1° = 60′ | 1′ = 1/60° ≈ 0.016667° | 360° = 21,600′ | 1′ ≈ 0.000291 rad

Arcseconds (″) - High-Precision Science

An arcsecond (symbol ″, also "arcsec" or "second of arc") is 1/60 of an arcminute, or 1/3,600 of a degree. Arcseconds are used in contexts demanding extreme precision. Astronomers measure stellar parallax and the positions of stars in arcseconds - even sub-arcsecond resolution for space telescopes. Geodesists use arcseconds in specifying geographic coordinates to centimetre-level accuracy. Satellite guidance systems and high-precision optical instruments routinely work at arcsecond and sub-arcsecond scales.

Key facts: 1° = 3,600″ | 1′ = 60″ | 1″ ≈ 0.000278° ≈ 4.848 × 10⁻⁶ rad

NATO Mils - Military Angle Measurement

The mil (NATO milliradian) divides a full circle into exactly 6,400 mils. It is widely used by NATO armed forces for artillery, mortar, and direct-fire targeting. The practical appeal of mils in ballistics comes from a useful approximation: at a range of 1,000 metres, 1 mil subtends approximately 1 metre. This makes range estimation and lateral corrections straightforward in the field without complex calculations. Different military traditions use slightly different mil definitions (Soviet/Russian artillery uses 6,000 mils per circle), so always confirm which standard applies.

Key facts: 6,400 NATO mils = 360° | 1,600 mils = 90° | 1 mil = 0.05625°

Milliradians (mrad) - Engineering & Optics

A milliradian (mrad) is simply one-thousandth of a radian. A full circle contains approximately 6,283.185 mrad (exactly 2,000π). Milliradians are commonly used in precision rifle scopes, long-range shooting, and optical engineering. In rifle scopes marketed as "MRAD," each click of adjustment typically corresponds to 0.1 mrad of angular change - at 100 metres that equals a 1 cm shift in point of impact. Milliradians also appear in radar systems, antenna beam-width specifications, and structural deflection measurements.

Key facts: 1 rad = 1,000 mrad | 1 mrad ≈ 0.05730° | 1 mrad ≈ 3.438′