Angle Converter
Convert between degrees, radians, gradians, arcminutes, arcseconds, and more angle units.
Quick Reference — 1 unit = ? Radian (rad)
| Unit | Value |
|---|---|
| Degree (°) | 0.01745329252 |
| Radian (rad) | 1 |
| Gradian (grad) | 0.01570796327 |
| Arcminute (′) | 0.0002908882087 |
| Arcsecond (″) | 0.000004848136811 |
| Turn / Revolution | 6.283185307 |
| Milliradian (mrad) | 0.001 |
Formula
Multiply by (from factor / to factor)
Everyday Uses
DIY slopes and cuts
A miter saw set in degrees, plans that say "1 in 12 slope," a protractor in radians class — convert between all of them.
Trigonometry homework
Calculators default to the wrong mode constantly — convert degrees ↔ radians and stop losing marks to it.
Navigation and mapping
Bearings in degrees, GPS in decimal, old charts in gradians — translate angles across navigation systems.
Ramp gradients
Accessibility specs quote slopes as ratios or percentages; builders think degrees — convert to meet code exactly.
Frequently Asked Questions
How do you convert between degrees and radians?
1 radian = 180/π ≈ 57.2958°. 1 degree = π/180 ≈ 0.017453 radians. To convert degrees to radians: multiply by π/180. To convert radians to degrees: multiply by 180/π. Key reference values: 30° = π/6 rad; 45° = π/4 rad; 60° = π/3 rad; 90° = π/2 rad; 180° = π rad; 270° = 3π/2 rad; 360° = 2π rad ≈ 6.2832 rad.
Why are radians used instead of degrees in mathematics and physics?
Radians make calculus and physics formulas cleaner. The derivative of sin(x) is cos(x) only when x is in radians — in degrees it would be (π/180)cos(x). Arc length formula: s = rθ only works with θ in radians. Angular velocity: ω = θ/t in rad/s, directly giving v = ωr. Most programming languages (sin, cos, tan in Python, JavaScript, C) expect radians. Degrees are more intuitive for everyday use; radians are natural for mathematics.
What is a gradian (grad) and where is it used?
A gradian (grad, or gon) divides a full circle into 400 parts, so a right angle = 100 grads. This makes it convenient: 1 grad = 0.9° = π/200 rad. Gradians are used primarily in surveying, civil engineering, and navigation in continental Europe — particularly in France, Germany, and the Netherlands. They simplify slope calculations (a 1% slope = 0.01 grads per metre of horizontal distance). Most scientific calculators include a "Grad" mode alongside Deg and Rad.
What are arcminutes and arcseconds used for?
1 degree = 60 arcminutes (′). 1 arcminute = 60 arcseconds (″). So 1 degree = 3,600 arcseconds. Arcminutes and arcseconds are used in: astronomy (the Moon's diameter ≈ 31 arcminutes; Jupiter at opposition ≈ 50 arcseconds); navigation and GPS coordinates (latitude/longitude in degrees, minutes, seconds — DMS format: 51°30′26″N); telescopes (resolving power: the Hubble Space Telescope resolves to 0.05 arcseconds); and rifle/artillery ballistics.
How does angular measurement work in navigation and GPS coordinates?
Geographic coordinates are expressed as degrees, minutes, seconds (DMS) or decimal degrees (DD). London: 51°30'26"N, 0°7'39"W (DMS) = 51.5072°N, 0.1275°W (decimal degrees). To convert DMS to decimal: DD = degrees + minutes/60 + seconds/3600. To convert decimal to DMS: the integer is degrees; multiply the decimal by 60 for minutes; multiply the remaining decimal by 60 for seconds. GPS systems typically output decimal degrees; maps often show DMS. 1 arcminute of latitude = 1 nautical mile = 1.852 km.