Angle 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 30В°, 45В°, 90В° 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.

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: 360В° = 2ПЂ radians = 400 gradians = 1 turn. Half that is 180В° = ПЂ rad = 200 grad. A right angle is 90В° = ПЂ/2 rad = 100 grad.

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 ПЂ/180 factor in every derivative. More to memorize, more chances for error.

In programming: JavaScript, Python, C++, and most other languages expect radians in trig functions. Math.sin(), Math.cos(), and Math.tan() all take radians. Math.atan2() returns radians. To convert, use radians = degrees * Math.PI / 180. Interfaces may show degrees for readability, but the underlying calculation often needs radians. Angle wrapping also matters when results must stay inside a range such as -180В° to 180В°.

When Angle Conversions Matter

Angle conversions appear in trigonometry, navigation, mapping, surveying, CAD, robotics, astronomy, and software development. The right unit depends on the context: degrees are easy to read, radians are natural for math, gradians are still used in some surveying workflows, and DMS notation is common in coordinates.

Trigonometry and Calculators

Most classroom problems show angles in degrees, while calculus and programming libraries usually use radians. When solving trig expressions, confirm the angle mode before using sine, cosine, tangent, inverse trig functions, or graphing tools. For triangle-side and trig function work, use the trig calculator after the angle unit is converted.

Navigation, Bearings, and Coordinates

Compass bearings are usually written in degrees from north. Latitude and longitude may appear as decimal degrees or DMS, such as 40 degrees 26 minutes 46 seconds. Converting DMS to decimal degrees makes coordinates easier to paste into maps, spreadsheets, and GIS tools.

Normalization and Rotation

An angle can describe the same direction in more than one way. For example, 450 degrees points the same direction as 90 degrees after one full rotation is removed. Normalizing to 0 to 360 degrees is common for compass-style directions, while -180 to 180 degrees is common in some programming and mapping workflows.

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