Click any row to load that angle into the converter.
How to Use This Converter
Pick your source and target units from the dropdowns, type in a value, and hit Convert. The swap button (↔) flips the conversion direction if you need to go the other way.
The Quick Conversion buttons set common unit pairs. The Preset Angles fill in values like 30°, 45°, 90° so you don't have to type them. If you want to see an angle in every unit at once, switch to Convert to All mode.
Negative angles convert correctly—they represent clockwise rotation. Values over 360° represent multiple rotations. Use the Options panel if you need to adjust decimal places, turn on scientific notation, or normalize angles to a specific range.
Quick Examples
45 degrees to radians:
What about π/6 radians to degrees? Multiply by 180/π: that gives you (π/6) × (180/π) = 180/6 = 30°.
A compass bearing of 225°:
Degrees, Radians, and Gradians
Degrees go back to the ancient Babylonians and their base-60 math. Why 360 in a circle? Maybe it was close to the days in a year, or maybe they just liked that 360 divides evenly by so many numbers (2, 3, 4, 5, 6, 8, 9, 10, 12...). Degrees stuck around because they work well in everyday situations—construction, navigation, giving directions.
Calculus and physics need something different. Radians define an angle by the arc length: one radian is when the arc equals the radius. A full circle is 2π radians (about 6.283). The derivative of sin(x) equals cos(x)—but only when x is in radians. Use degrees and you'd have π/180 factors everywhere.
Gradians? French metric system, 1790s. A right angle is exactly 100 gradians, full circle is 400. Surveyors use them—slope calculations come out as clean percentages. You'll still see a GRAD mode on scientific calculators.
Key Conversion Formulas
Here's how they relate: 360° = 2π radians = 400 gradians = 1 turn. Half that is 180° = π rad = 200 grad. A right angle is 90° = π/2 rad = 100 grad.
deg = rad × 180/π
(Memorize: π rad = 180°)
Gradians to degrees: multiply by 0.9
For quick estimates: 1° ≈ 0.0175 rad, and 1 rad ≈ 57.3°
Arcminutes, Arcseconds, and DMS
One degree = 60 arcminutes (symbol: ′). One arcminute = 60 arcseconds (symbol: ″). So there are 3,600 arcseconds in a degree.
Astronomers measure star positions in arcseconds—the Moon spans about 31 arcminutes across, roughly half a degree. Rifle scopes use MOA (minute of angle) for adjustments. GPS coordinates and surveyors sometimes write angles in DMS notation: 45° 30′ 15″ instead of 45.504167°.
Why Radians Matter in Math
Radians come directly from circle geometry—the arc length divided by the radius. That's not arbitrary. Calculus relies on this: d/dx sin(x) = cos(x), but only when x is in radians. Same for d/dx cos(x) = -sin(x). The small angle approximation (sin θ ≈ θ for small θ) also requires radians. And arc length simplifies to s = rθ.
Degrees would require a π/180 factor in every derivative. More to memorize, more chances for error.
In programming: JavaScript, Python, C++, and basically every other language expect radians in their trig functions. Math.sin(), Math.cos(), and Math.tan() all take radians. Math.atan2() returns radians. To convert, use radians = degrees * Math.PI / 180. Show angles to users in degrees since that's what people understand, but run your calculations in radians since that's what the library expects. If you need angles in a specific range like -180° to 180°, watch out for wrapping issues.