Free online angle converter. Convert degrees to radians, gradians, arcminutes, arcseconds, and turns instantly — for trigonometry, engineering, CNC machining, and navigation. Accurate, instant, no sign-up.
degrees • radians • gradians • arcmin • arcsec
Convert degrees to radians for sin, cos, tan in calculus, where radians are required for derivatives and series.
JavaScript, Python, and C use radians in Math.sin(). CNC and CAD often use degrees. Convert to avoid errors.
Telescopes and star positions use arcminutes and arcseconds — 1 degree = 60 arcmin = 3,600 arcsec.
Angles can be measured in several units, and the most common conversion is between degrees and radians. A full circle is 360 degrees, 2π radians (approximately 6.2832), 400 gradians, or 1 turn. The key relationship: 1 radian = 180/π ≈ 57.2958 degrees, and 1 degree = π/180 ≈ 0.01745 radians. Degrees are intuitive for everyday use (a right angle is 90°), but radians are the natural unit in higher mathematics because they make calculus formulas clean — the derivative of sin(x) is cos(x) only when x is in radians.
Common angles convert to memorable radian values: 30° = π/6, 45° = π/4, 60° = π/3, 90° = π/2, 180° = π, and 360° = 2π. Gradians (also called gons) divide a right angle into 100 parts and are used in surveying and some European engineering. For precise work in astronomy and navigation, degrees are subdivided into arcminutes (1° = 60′) and arcseconds (1′ = 60″), so 1 degree = 3,600 arcseconds. Milliradians (mrad) are used in ballistics and optics for fine angular adjustments.
30° = 0.5236 rad = π/6. 45° = 0.7854 rad = π/4. 60° = 1.0472 rad = π/3. 90° = 1.5708 rad = π/2. 180° = 3.1416 rad = π.
Radians are dimensionless (arc length ÷ radius), making trig derivatives and Taylor series work cleanly. d/dx sin(x) = cos(x) requires radians. This is why programming Math functions use radians.
1 gradian = 1/400 of a circle = 0.9°. A right angle = 100 gon. Used in continental European surveying and some calculators (GRAD mode).
1° = 60 arcmin = 3,600 arcsec. The full Moon spans ~30 arcmin. Hubble resolves ~0.05 arcsec. Star coordinates use arcsec precision.
The degree is a historical accident that has outlived every attempt to replace it: 360 comes from Babylonian base-60 arithmetic and the convenient fact that 360 divides evenly by 24 different numbers. The radian, by contrast, is the mathematically natural unit — one radian is the angle subtending an arc equal to the circle's radius, so a full turn is exactly 2π radians. That definition is what makes calculus work cleanly on trigonometric functions, and it is why every programming language's sin() and cos() expect radians. Passing degrees to them is one of the most common numerical bugs there is.
radians = degrees × π ÷ 180degrees = radians × 180 ÷ πgradians = degrees × 10 ÷ 91 degree = 60 arcminutes = 3,600 arcsecondswhere:
Assumptions: Trigonometric functions in virtually every programming language take radians. Spreadsheet formulas do too — Excel's SIN() needs RADIANS() wrapped around a degree value.
Convert a right angle into the two alternative systems.
Result90° = 1.5708 radians = 100 gradians = 5,400 arcminutes
The useful constant is 1 radian = 57.29578°. If a trigonometric result looks plausible but wrong, checking the angle mode is almost always the first thing to try — the failure is silent because both inputs are valid numbers.