Angle Converter
Convert between degrees, radians, gradians, arcminutes, arcseconds, and more angle units.
Each step along the axis is ten times the one before — the angle units span 6 orders of magnitude, so a straight scale would bunch almost all of them at one end.
Instant insight
Degrees, radians and gradians all divide the circle differently
A full turn is 360°, 2π radians, or 400 gradians. Calculators silently return nonsense when set to the wrong mode.
Check the mode before trusting a trig result: sin(30°) must be exactly 0.5.
Check the direction of the conversion
Going from degree to radian, the number should get bigger if the new unit is smaller, and smaller if it is bigger.
If it moved the wrong way, the factor has been applied upside down — the most common conversion error there is.
Worked out in your browser — nothing is sent anywhere. Comparison figures are approximate.
Common conversions
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.
Cutting mitres
Convert between degrees, radians and gradians when a plan or saw uses a different unit from your protractor.
Bearings and navigation
Convert a compass bearing into radians for a calculation, or back again for the field.
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.