Angle Converter
Convert degrees, radians, gradians and arc units.
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About Angle
Angle covers five units: degrees, radians, gradians, arcminutes and arcseconds. Degrees are the everyday unit almost everyone learns first and the one used in navigation, construction, design software and geometry class. Radians are the mathematical unit, the one trigonometry, calculus and nearly every programming language expect. Gradians are a metric-style angle unit that survives in some surveying work. Arcminutes and arcseconds subdivide the degree into much finer parts and are used where tiny angles matter, notably astronomy, optics and map coordinates. The people who land here are usually programming, doing maths homework or reading a coordinate. The pages used most are degrees to radians, radians to degrees, degrees to arcminutes and degrees to gradians.
The practical thing to understand is why radians exist at all, because degrees seem perfectly good until they suddenly are not. The degree is an arbitrary division of a circle inherited from ancient astronomy; there is no mathematical reason a full turn should be split that particular way. The radian instead comes out of the geometry of the circle itself, which makes the calculus of trigonometric functions clean and is exactly why mathematics, physics and software adopted it. This causes the single most common bug in this whole category: trigonometric functions in most programming languages take radians, not degrees. Passing a degree value straight into a sine or cosine call compiles happily and returns a plausible-looking number that is completely wrong, with no error to warn you. If a rotation, an animation or a geometry calculation is behaving strangely, checking the units is usually the first thing to try. Arcminutes and arcseconds carry a second, smaller trap: they subdivide the degree the way minutes and seconds subdivide an hour, not in tenths, so a coordinate written in degrees, minutes and seconds is not a decimal number and cannot be read as one.
Every pair page here shows the formula it used, walks through a worked example, and gives a reference table covering the angles people look up most, such as the quarter and half turns. All of the pages are free, require no account, and run entirely in your browser, so nothing you enter is uploaded anywhere.
Frequently asked questions
Why do programming languages use radians instead of degrees?
Because the radian comes from the geometry of the circle itself rather than from an arbitrary division of it, which makes the mathematics of trigonometric functions much cleaner. Nearly every language follows the mathematical convention, so sine, cosine and tangent expect radians. Passing degrees in gives a wrong answer with no error message at all, which makes it one of the easiest bugs to introduce and one of the hardest to spot.
What are arcminutes and arcseconds?
They are subdivisions of a degree, used where angles get very small, such as in astronomy, optics and geographic coordinates. They work like minutes and seconds of an hour rather than like decimals, so an arcminute is a fraction of a degree and an arcsecond is a fraction of an arcminute. That matters when reading a coordinate written in degrees, minutes and seconds, because it is not a decimal number and cannot be treated as one.
What is a gradian and does anyone still use it?
The gradian is an angle unit designed to fit the metric style, dividing a right angle into a round hundred parts. It never displaced the degree in general use, but it survives in some surveying and civil engineering practice, particularly in parts of Europe, and it appears as a mode on many scientific calculators. Most people meet it only by pressing the wrong calculator button, which is worth checking if your trig answers look wrong.
Do I need an account to use these pages?
No. Every angle conversion page is free, requires no sign-up and has no usage limit. The calculation is done in your browser with JavaScript, so nothing you enter is sent to a server. Each page also includes a reference table of common angles, which is often faster than typing when you want a standard value.