What is Angle Conversion?
Convert radians, degrees, gradians, and more.
Supported units: Radian (rad), Degree (°), Gradian (grad).
Why the circle has 360 degrees (and three competing ways to measure it)
The degree comes from Babylonian astronomy: they used base-60 arithmetic and chose 360 parts per circle because it's close to the number of days in a year and divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 45, 60, 72, 90, 120, 180 — extremely convenient for calendar, navigation, and astronomical work. The radian is the "natural" mathematical unit: 1 radian is the angle that cuts off an arc equal to the radius. Because circumference = 2πr, a full circle = 2π radians. This makes calculus clean — the derivative of sin(x) equals cos(x) only when x is in radians. The gradian (or gon) was invented during the French metric reform in the 1790s: 400 gradians per circle, so 100 per quadrant. It never caught on except in surveying in a few European countries. Every programming language's trig functions (Math.sin, sin()…) expect radians, not degrees, which is a source of endless first-year student bugs.
When you need to convert angles
Navigation and compasses
Bearings are reported in degrees (0°–360°, with north = 0). GPS devices sometimes output in degrees, minutes, seconds (DMS: 40°42′46″N), sometimes in decimal degrees (40.7127°) — converting between the two is an everyday task for anyone doing outdoor work.
Math and physics
All trigonometric identities, Taylor series, and physics equations assume radians. Converting degrees → radians is one of the first unit checks when plugging numbers into a formula from a textbook.
CAD and engineering
Mechanical drawings use degrees for tolerances (e.g. 45° ±0.5°); CNC machine G-code accepts either degrees or radians depending on the controller. Mixing the two at the wrong moment produces parts off by a factor of 57.3×.
Programming and graphics
JavaScript, Python, Java, C++ — all their trig functions take radians. Rotating a game sprite by "45°" actually means Math.sin(45 * Math.PI / 180). Libraries that "accept degrees" are wrappers around this exact conversion.
Surveying and GIS
Land surveys use DMS or decimal degrees, plus bearings expressed as compass directions ("N 32° 15′ E"). Maps, national grid coordinates, and satellite imagery all require fluent conversion across these forms.
Common Conversions
- 1 Radian (rad) = 57.2958 Degree (°)
- 1 Radian (rad) = 63.662 Gradian (grad)
- 1 Degree (°) = 0.0174533 Radian (rad)
- 1 Degree (°) = 1.11111 Gradian (grad)
- 1 Gradian (grad) = 0.015708 Radian (rad)
- 1 Gradian (grad) = 0.9 Degree (°)
FAQ
Q: How many degrees in 1 radian?
A: 1 radian ≈ 57.2958 degrees.
Q: How to convert degrees to radians?
A: Multiply by π/180. Example: 180° × π/180 = π rad.
Q: What is a gradian?
A: A gradian (grad) is 1/400 of a full circle, so 1 grad = 0.9 degrees.
Q: Why do math and programming use radians instead of degrees?
A: Because calculus works cleanly only in radians. The derivative of sin(x) is cos(x) in radians, but (π/180)·cos(x) in degrees — an awkward constant nobody wants to carry around. Every calculation involving rates of change of angles (signal processing, rotation, waves) is simpler in radians.
Q: What does DMS notation mean?
A: DMS = degrees, minutes, seconds. 1 degree = 60 minutes, 1 minute = 60 seconds (same Babylonian base-60 roots as the clock). "40°42′46″" means 40 degrees + 42 minutes + 46 seconds = 40 + 42/60 + 46/3600 ≈ 40.7128°. GPS coordinates, map grids, and astronomy all use this notation.