Convert between degrees, radians, gradians, and more angular measurements
Right angle (90°)
An angle is formed when two rays share a common endpoint, called the vertex. Angles are fundamental in mathematics, physics, engineering, navigation, and many other fields. Different measurement systems have evolved for different applications and cultural contexts.
Degrees (°) - The most common unit in everyday use, where a full circle equals 360°. This system dates back to ancient Babylonian astronomy, possibly chosen because 360 is divisible by many numbers.
Radians (rad) - The mathematical standard unit where a full circle equals 2π radians. One radian is the angle subtended by an arc equal in length to the radius. This unit makes calculus and trigonometry more elegant.
Gradians (gon) - A decimal-based system where a full circle equals 400 gradians. Developed during the French Revolution as part of metrification, it's still used in some surveying applications.
Turns or Revolutions - Simply counting complete rotations. One turn equals one complete circle (360° or 2π radians). Very intuitive for mechanical applications.
Mils - Military and artillery unit where a full circle equals 6400 mils (NATO standard). Originally based on the milliradian, it provides precise angular measurements for targeting and navigation.
Arcminutes and Arcseconds - Subdivisions of degrees used in astronomy, surveying, and precision applications:
Navigation & GPS: Ships and aircraft use angular measurements for course headings and position calculations. GPS coordinates use degrees, minutes, and seconds.
Engineering & Manufacturing: Mechanical parts often require precise angular tolerances. Rotating machinery depends on angular velocity measurements.
Astronomy: Telescope pointing and star cataloging require extremely precise angular measurements, often to arcsecond accuracy.
Artillery & Surveying: Military targeting systems use mils for precision, while land surveying relies on degrees and their subdivisions.
Computer Graphics: 3D modeling and game engines typically use radians for rotations internally, though degrees are shown to users.
90° = π/2 rad = 100 gon
180° = π rad = 200 gon
360° = 2π rad = 400 gon
π rad ≈ 180° (half circle)
2π rad ≈ 360° (full circle)
1 rad ≈ 57.3°
1° = 60' = 3600"
1' ≈ 0.3 mm at 1 km
1" ≈ 5 μm at 1 km
Degrees: General use
Radians: Mathematics
Mils: Military/precise targeting
The degree system originated with ancient Babylonian astronomers who used a base-60 numbering system. They approximated the year as 360 days, leading to 360 degrees in a circle. The radian was introduced much later (early 1800s) as mathematics became more sophisticated. The gradian system was part of the French Revolutionary attempt to decimalize all measurements, though it never gained widespread adoption outside of surveying.