Free Angle Converter
Convert angle units instantly — degrees, radians, gradians, arcminutes, arcseconds, and turns. Free, symmetrical, no sign-up.
Exact Conversion Formula & Multipliers
Angle conversion runs on the relationship between a full rotation expressed three ways: 360 degrees = 2π radians ≈ 6.283185 radians = 400 gradians. So 1 radian = 180/π degrees ≈ 57.29578°, and 1 gradian = 0.9°. Because π is irrational, degree-to-radian conversions never terminate exactly in decimal form, which is why this converter carries results to enough decimal places to avoid visible rounding drift in downstream calculations.
Practical Real-World Scenarios
- Engineering and physics: rotational mechanics, wave functions, and calculus-based formulas are written in radians because they simplify derivative and integral calculations.
- Navigation and surveying: compass bearings and land surveys are traditionally expressed in degrees, and some European surveying traditions use gradians instead.
- Programming and CAD software: most programming languages' trigonometric functions expect radians as input, while user interfaces typically display degrees, requiring a conversion step in the code.
Historical & Technical Background
The 360-degree circle likely originates from ancient Babylonian astronomy, where a base-60 number system combined with a roughly 360-day approximation of the solar year made 360 a convenient, highly divisible number for dividing a circle. The radian, defined by the ratio of arc length to radius, is comparatively recent — formalized in the 19th century — and was designed specifically to make calculus with angles mathematically clean rather than to match any calendar or counting convention. The gradian, a French Revolutionary-era metric attempt at decimalizing angles, never achieved the same adoption as the meter or gram.
Precision & Accuracy Statement
Conversion factors use the exact defined values given in NIST Special Publication 811 and the BIPM SI Brochure wherever such a definition exists (for example 1 lb = 0.45359237 kg, 1 in = 2.54 cm, 1 US gal = 3.785411784 L, g₀ = 9.80665 m/s²); derived factors are computed from those definitions rather than copied as rounded figures. Results are calculated in double-precision floating point and shown to 10 significant digits, so the last digit can differ by one unit of rounding. The rounded figures in the text and tables above are for reading convenience.
How This Converter Works
Degrees are used in everyday life, while radians are essential for mathematics and physics. Gradians are used in surveying, and arcminutes/arcseconds in astronomy and navigation. All 6 units update simultaneously.
Common Conversions
| Degrees (°) | Radians (rad) | Gradians (grad) | Context |
|---|---|---|---|
| 30° | 0.5236 rad | 33.33 grad | Half the interior angle of an equilateral triangle |
| 45° | 0.7854 rad | 50 grad | Diagonal, 1/8 of full rotation |
| 90° | 1.5708 rad | 100 grad | Right angle |
| 180° | 3.1416 rad | 200 grad | Straight line, π radians |
| 360° | 6.2832 rad | 400 grad | Full rotation, 2π radians |