Angle Converter

Six units, one turn

An angle measures how far something has rotated, and the units are just different ways of slicing the same full circle. The degree, the oldest and most familiar, comes from Babylonian astronomy: astronomers counted in sixties, and a circle of 360 parts divides cleanly by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more, which made hand calculation practical on clay tablets. Nothing in geometry forces the number 360. It survived because it is convenient.

The radian takes a different approach. Instead of slicing the circle into an arbitrary count, it asks how long the arc is compared with the radius. Slide the radius around the rim and it wraps 6.2832 times, written 2 pi, so a full turn is 2 pi radians. The NIST Special Publication 811, the United States guide to SI units, lists the radian as the SI derived unit for plane angle, and every formula in calculus and physics assumes it.

The gradian is the metric reform attempt: French surveyors in the 1790s split the right angle into 100 parts so that a full turn is 400. It still runs on the display of most theodolites sold in Europe and South America. The turn, sometimes called a revolution or a rotation, is what machinists and programmers mean by rev or rotation. Arcminutes and arcseconds split each degree into 60 and then 60 again, and the NATO mil slices the turn into 6,400 parts for artillery work.

All six describe the same quantity, so conversion is always a single multiplication. The factors are fixed constants, and the table in the next section lists them once so you never have to trust a chained series of lookups.

The factors, and how to apply them

Convert by going through degrees or straight to the target. One degree holds 60 arcminutes, each arcminute holds 60 arcseconds, so a full degree contains 60 × 60 = 3,600 arcseconds. One radian is 57.2958 degrees, one gradian is 0.9 degrees, one turn is 360 degrees and one NATO mil is 360 ÷ 6400 = 0.05625 degrees.

Two worked examples cover the common directions. To turn 12.5 degrees into radians, multiply by pi and divide by 180: 12.5 × 3.14159 ÷ 180 = 0.2182 rad. To go the other way, 2.5 radians is 2.5 × 57.2958 = 143.2395 degrees. The BIPM, the treaty organisation that maintains the SI, publishes the same factors in its SI brochure, because the radian and the degree are both defined against one and the same full turn.

Keep the order of operations straight when chaining: multiply or divide first, and round only at the end. Rounding 0.2182 to 0.22 before a later step doubles the error, while rounding the final display changes nothing downstream.

Radians are the unit that math and code expect

Every programming language takes radians. JavaScript's Math.sin, Python's math.sin, Excel's SIN and the trig keys on a scientific calculator in RAD mode all assume that a full turn is 2 pi. Feed a function 90 expecting a right angle and you get the sine of 1.5708 radians, which happens to be almost exactly 1, and a coincidence that hides the mistake. Feed it 30 expecting degrees and the answer is nowhere near 0.5.

The deeper reason is that radians make formulas shorter. Arc length is radius times the angle: a radius of 5 m swept through 2 rad cuts an arc of 5 × 2 = 10 m. In degrees the same formula would need a factor of pi ÷ 180 bolted onto every line. The derivative of the sine is the cosine only in radians, and the small angle approximation, where the sine of a small angle is close to the angle itself, is exact only in radians too.

One degree is 0.01745 rad, so a quick sanity check works like this: results below 1.5708 are angles within a quarter turn, and a result like 0.7854 is the radian face of 45 degrees. When a drawing gives you degrees and the code wants radians, convert once at the input and keep the rest of the program in radians throughout. Mixing the two inside one calculation is the single most common trigonometry bug in game and graphics code.

Gradians, mils and turns: who uses them

Gradians survive because of one piece of mental arithmetic: a right angle is 100 grad, so quarter angles are round numbers and perpendicularity checks on a construction site need no conversion. A slope read as 50 grad is exactly 45 degrees. Modern digital theodolites display either degrees or gradians with a menu setting, and the instrument converts internally, but site drawings in gradian countries are still drawn in gradians, so the two systems meet on every imported plan.

The mil comes from artillery, where the useful fact is that a small angle subtends a width proportional to range. NATO standardised the turn into 6,400 mils, which puts one mil at 0.05625 degrees and just under 1 metre of width per kilometre of range. That near miss is the point: soldiers range with the rounded rule and the optics are calibrated in the exact unit. Beware the other mil: the milliradian system divides the turn into 6,283.19 parts, and one milliradian is 0.0573 degrees. Hunting scopes sold in MRAD and military maps in NATO mils differ by about 2 percent, enough to miss a target at long range.

Turns are the unit of rotation in software. CSS animations, motor controllers and CNC spindle speeds all count revolutions, and converting to degrees is a single multiplication by 360. A stepper motor spec of 200 steps per turn is 1.8 degrees per step, and half stepping halves that to 0.9 degrees.

Arcminutes and arcseconds: when a degree is too coarse

A degree is a blunt instrument for anything seen through a telescope or aimed down a barrel. Splitting it into 60 arcminutes and each arcminute into 60 arcseconds gives 3,600 arcseconds per degree, fine enough to describe the resolution of large observatories, which reach a few hundredths of an arcsecond after correcting for the atmosphere.

The scales show up in daily estimates more often than people notice. The full Moon spans about 31 arcminutes, which is 31 ÷ 60 = 0.5167 degrees, and the Sun is nearly the same size in the sky, which is why total solar eclipses work. Human visual acuity is conventionally 1 arcminute, the basis of the 20/20 line on an eye chart. In navigation, one arcminute of latitude along a great circle defines the nautical mile, fixed at exactly 1,852 metres internationally.

Shooting culture abbreviates arcminute to MOA. One MOA spans about 1.047 inches at 100 yards, so a rifle that groups inside 1 MOA holds roughly a 1 inch group at that distance, and the group grows in proportion to range. Scope turrets click in fractions of an MOA, usually a quarter, which moves the point of impact about 0.26 inches per click at 100 yards.

Common angles in every unit

The angles below cover most drawings, code reviews and shop floor conversations. Each row is the same rotation written five ways, and the pattern is worth a glance: gradians always read 10 divided by 9 of the degrees, turns are degrees divided by 360, and the radians of 30, 45 and 60 degrees are pi divided by 6, by 4 and by 3.

The 30, 45 and 60 degree rows come from triangles with sides in whole number and root ratios, which is why they appear in every trigonometry table and why the reference angle calculator leans on them. An angle outside 0 to 360 converts the same way; a coterminal angle such as 390 degrees carries the same radian value as 30 degrees, just past one full turn.

Rounding, and the calculator mode trap

Two habits prevent almost every angle conversion error. The first is to carry full precision through the work and round only the displayed result. Rounding pi to 3.14 costs about 0.05 percent on a degree to radian conversion, small on one step and compounding across a chain of them. Four decimals is plenty for construction and code, and the difference between 1.5708 and 1.570796 is far below what any drawing can show.

The second is to know which mode the calculator is in before trusting a result. Convert 90 degrees in DEG mode with a degree to radian function and you get 1.5708. Take the sine of 90 in RAD mode and the calculator reads the input as 90 radians and answers 0.8940, which is a completely different question. The tell is magnitude: a plain angle in radians must sit below 6.2832, and a plain angle in degrees beyond 360 means the input was in the wrong unit to begin with.

Anchor values make the check instant. One radian is 57.2958 degrees, a right angle is 1.5708 rad or 100 grad, a full turn is 6.2832 rad, 400 grad or 6,400 mils. If a converted value lands near one of those anchors in the wrong place, something inverted along the way. Multiplying when you should have divided produces a number off by a factor of about 3,283 between degrees and radians, which is far too large to miss.

Doing the conversion in your head

Three tricks cover the directions that come up away from a desk. Degrees to gradians: multiply by 10 and divide by 9, so 45 × 10 ÷ 9 = 50 grad. Gradians back to degrees: multiply by 9 and divide by 10. Neither ever needs pi.

Degrees to radians: multiply by 0.01745, which is pi over 180 rounded, or divide by 57.3 when the angle is large. Thirty degrees is close to half of 57.3, so the answer lands a bit above 0.5, and 0.5236 is right. Radians to degrees: multiply by 57.3, and correct toward 57.2958 when the angle matters. Two radians is about 114.6 degrees, three is about 171.9, and four crosses the half turn.

For the fine scales, divide or multiply by 60 one step at a time instead of by 3,600 in one jump. Half a degree is 30 arcminutes, and a quarter of an arcminute is 15 arcseconds. Artillery and long range shooting use the proportional rule directly: an angle of 10 mils spans about 10 units of width per 1,000 units of range, so 10 mils at 800 m is close to 8 m of drift. The exact NATO figure is 0.982 m per mil per kilometre, and the rounded rule is deliberately generous by about 2 percent.