Angle Converter

Introduction

Angles are one of the few quantities in physics and engineering where a single physical measurement, say, the rotation of a wheel, the bearing of a road, or the latitude of a city, can be written down in at least seven different unit systems. Pick the wrong unit and the math goes silently wrong: a sine function expects radians, a survey record expects degrees-minutes-seconds, an artillery scope reads NATO mils, a CNC machine may output gradians, and a paper map labels coordinates in degrees with a decimal-minutes suffix. This Angle Converter exists so that no matter which unit you start with, you can see the equivalent value in every other common unit at the same time, and round-trip between degrees-minutes-seconds and decimal degrees without losing a second of arc.

The converter is built on a single canonical base, the turn (one full revolution). Every other unit is defined as an exact rational multiple or exact irrational multiple of the turn. Because the turn is dimensionless, it counts how many times you go around, the conversion arithmetic never accumulates drift between the integer-multiple units: degrees, gradians, arcminutes, arcseconds and NATO mils round-trip exactly. Radians and milliradians carry a factor of π and so round to the precision you ask for.

How to Use

1. Type a number into the Value field on the left of the top panel. You can use decimal notation, and you can paste values like 1.5708 or 90 or even 3.14159. Negative values are accepted and represent clockwise or negative rotations depending on context.
2. Pick the source unit from the From unit dropdown. The dropdown lists every unit the converter understands: degrees, radians, gradians (gons), turns, arcminutes, arcseconds, true milliradians, and NATO mils.
3. Choose the Significant figures you want. The converter offers 4, 6, or 8 significant digits. 4 is plenty for homework and rough sketching; 6 is the default for engineering work; 8 covers surveying and geodesy.
4. Read the live results table below. Every other unit appears in its own row, formatted to the chosen significant-figure precision. The input unit is marked with a small "(input)" tag.
5. Click any preset button to load a common angle. The presets are right angle (90°), straight (180°), full turn (360°), π/4, π/3, 1 radian, 100 gradians (one metric right angle), and 1 arcsecond.
6. Use the DMS panel for GPS-style coordinates. Type a DMS value into the left box (for example 45°30'30"), and the right box updates with the decimal-degree equivalent. Type a decimal degree and the left box updates back to DMS.

The Unit Definitions

The radian is the SI-coherent unit for angle. It is defined as the ratio of arc length to radius: an angle of 1 radian sweeps out an arc whose length equals the radius of the circle. Because it is a ratio of two lengths, the radian is dimensionless, pure number, no units attached in SI, and it is the only angle unit for which the standard series expansions of sine, cosine and tangent work without an extra conversion factor. That is why Math.sin in JavaScript, sin in Python's math module, np.sin in NumPy and sin in spreadsheets all expect radians. Anything else needs a * π / 180 or / (180 / π) wrapper.

The degree is 1/360 of a turn, a convention inherited from Babylonian sexagesimal counting. Babylonians divided the circle into 360 parts because 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120 and 180, handy for everyday fractions. There is nothing physically special about 360; the French revolutionaries briefly tried to replace it with 400 gradians as part of the metric system, but degrees had too much momentum.

The gradian, also called the gon, divides the right angle into 100 equal parts, so a full turn is exactly 400 gradians. This makes right-angle arithmetic trivial: 25% of a right angle is 25 gradians, 50% is 50 gradians, and a 30% slope reads as a clean number. Some European surveying instruments still use gradians, and a few CNC controllers use them as the default angular unit, so it is worth recognising.

The turn (also revolution or cycle) is the most physically obvious unit: one full rotation. Astronomers and physicists like it because it makes fractions transparent: a quarter turn is 90°, a half turn is 180°, two-thirds of a turn is 240°. Angular velocity in turns per second is also immediately intuitive.

Arcminutes and arcseconds are the sexagesimal subdivisions of the degree: 1 degree = 60 arcminutes, 1 arcminute = 60 arcseconds. A full turn is therefore 21,600 arcminutes or 1,296,000 arcseconds. They are the historical units of astronomy and celestial navigation and remain the standard format for geographic coordinates in printed maps and many handheld GPS units.

The milliradian is 1/1000 of a radian, so a full turn is exactly 2000π milliradians, or approximately 6283.185 milliradians. It is the angular unit of choice for rifle scope reticles and some other long-range aiming applications, because a target one metre across at a range of one thousand metres subtends almost exactly 1 milliradian.

The NATO mil, often written with a small "m", is a separate unit defined as 1/6400 of a turn. It is the angular unit of artillery sights, compasses, and some military surveying equipment. The deliberate choice of 6400 (rather than the true 2π × 1000 ≈ 6283.185) lets users do mil arithmetic in their head: a full turn is 6400, a half turn is 3200, a quarter is 1600. It is also noticeably different from the true milliradian, 1 NATO mil ≈ 1.0186 true milliradians, which is one of the easiest mistakes to make in ballistics problems.

Degrees-minutes-seconds (DMS) notation writes an angle as three sexagesimal components separated by the degree, arcminute and arcsecond symbols: 45°30'30". The arcminute symbol is a prime (′) and the arcsecond symbol is a double prime (″). DMS is the coordinate format used on paper charts, in surveying field books, and on many consumer GPS displays, and the most common request for an angle converter is to turn DMS coordinates into decimal degrees for a database or web map.

Worked Examples

Right angle to radians. A right angle is 90 degrees. Convert to radians using the factor π/180: 90 × π / 180 = π/2 ≈ 1.5707963 rad. This is the most-memorised conversion in trigonometry because the sine, cosine and tangent of π/2 are the clean values 1, 0 and undefined.

180 degrees to π. 180° = π radians ≈ 3.14159265 rad. This is the conversion that turns the half-circle into the fundamental constant of trigonometry. The converter returns the symbol π when the angle rounds cleanly to it, but the displayed numeric value is always available.

45°30'30" to decimal degrees. 45 degrees plus 30 arcminutes plus 30 arcseconds = 45 + 30/60 + 30/3600 = 45 + 0.5 + 0.008333… = 45.508333°. This is a real GPS latitude, within a few hundred metres of the geographic centre of France. The DMS panel of the converter handles this directly.

Radian to degree. 1 radian = 180/π ≈ 57.2957795°. Multiply by this factor to convert any radian value to degrees; divide by it to convert degrees to radians. The number is irrational and so cannot be written as an exact decimal, which is why significant-figure control matters.

NATO mils to degrees. 1 NATO mil = 360/6400 = 0.05625°. So 1600 NATO mils = 90° exactly. To convert the other way, multiply degrees by 6400/360 = 17.777…, note the recurring decimal.

True milliradian to NATO mil. 1 true milliradian ≈ 0.0572958°, while 1 NATO mil = 0.05625°. The ratio is 1.0186, so a target sighted at 10 NATO mils sits at 10.186 true milliradians.

Where It Shows Up

GPS and surveying. Consumer GPS units commonly display latitude and longitude in DMS, but most mapping software stores them as decimal degrees. Surveying instruments read bearings in degrees with minute-and-second precision, often written as N 45°30'30" E. Converting between the formats cleanly is the bread-and-butter task for an angle converter.

Trigonometry and the classroom. Every sine, cosine and tangent problem in a textbook starts by either telling you the angle in degrees and expecting you to convert to radians, or giving the angle in radians and expecting you to recognise a familiar value like π/4 or π/6. A quick sanity check across all unit systems is the easiest way to catch a misplaced factor of π/180.

CNC machining. Some CNC controllers, particularly European ones, accept or output angles in gradians rather than degrees. A 90° tool change might appear as 100 gradians in the G-code, and confusing the two wastes a setup cycle.

Astronomy. Right ascension is commonly given in hours-minutes-seconds of arc, declination in degrees-minutes-seconds of arc, and angular diameter in arcminutes or arcseconds. Going between any two of these is a routine calculation.

Artillery mils. A 155 mm howitzer's panoramic telescope is graduated in NATO mils (6400 per turn). Range tables are built around mils, but firing data is often re-expressed in degrees for cross-checking with civilian maps.

CAD and engineering drawing. Most CAD systems use degrees internally, but some legacy drafting standards use gradians, and 3D printing slicers may expose rotation in revolutions (turns) for clarity.

Common Mistakes

Degrees vs radians in code. The single most common angle bug in software is passing a degree value to Math.sin, Math.cos, np.sin, sin in spreadsheets or sin in calculators expecting radians. Math.sin(90) returns 0.8939966636, not 1, because 90 radians is many full turns past the right-angle peak. Always check the unit the function expects, and multiply by π/180 (or divide by it) at the boundary.

DMS decimal-minutes vs decimal-seconds. GPS units increasingly display coordinates in decimal minutes, e.g. 45° 30.5083'. The .5083 is minutes, not degrees, so the decimal-degree conversion is 45 + 30.5083/60 = 45.508472°, not 45.5083°. A coordinate written in decimal seconds (rare but used in some astronomical catalogues) splits the decimal at the seconds level instead, so 45° 30' 30.5" becomes 45 + 30/60 + 30.5/3600 = 45.508472°. The two formats share the same final decimal degrees by coincidence in some cases, which makes the confusion hard to catch.

NATO mil vs true milliradian. Both are called "mil" by different communities, and both are roughly the angular size of a one-metre target at one kilometre, but they are not the same unit. NATO mils are 1/6400 of a turn; true milliradians are 1/(2000π) of a turn. A 5% error at long range is the difference between a hit and a wide miss, so always check which "mil" the source actually means.

Mixing degrees and gradians in a single calculation. A right angle in gradians is exactly 100; a right angle in degrees is exactly 90. Adding them together gives an angle that is neither, and most calculators will silently return the wrong answer because they treat both as unitless numbers. The converter's unit dropdown is a useful discipline: pick one and stay in it.

Forgetting the sign of the result. DMS coordinates in the southern or western hemispheres are written with a trailing S or W letter rather than a leading minus sign: 45°30'30"S means −45.508333°. The converter's DMS parser strips both the letter and an explicit minus sign and returns a signed decimal, but a hand calculation that ignores the letter will silently flip the hemisphere.

Frequently Asked Questions

Why does Math.sin want radians?

Because the radian is the only angle unit for which the standard series expansions of the trigonometric functions, and the geometric derivations of the derivative identities, work without a conversion factor. Degrees and gradians both carry a scaling constant that has to be inserted and removed each time the function is called. Choosing radians as the convention keeps the constant implicit inside the function and lets the rest of the math stay clean. If your input is in degrees, multiply by π/180 before calling sin, and multiply the result by 180/π if you want degrees back.

What is the difference between a milliradian and a NATO mil?

A true milliradian is exactly 1/(2000π) of a turn, or about 0.0572958°. A NATO mil is exactly 1/6400 of a turn, or 0.05625°. They are within 2% of each other, which is why the two communities can use the same word "mil" without realising they disagree, but at artillery ranges the difference is significant. If you see "mil" in a ballistics context, assume NATO mils. If you see it on a rifle scope reticle, assume true milliradians unless the manual says otherwise.

How do I convert 45°30'30" to decimal degrees?

45 degrees plus 30 arcminutes plus 30 arcseconds. Convert minutes to a fraction of a degree by dividing by 60, and seconds by dividing by 3,600: 45 + 30/60 + 30/3600 = 45.508333°. The DMS panel of this converter does the same calculation automatically in both directions, and it correctly handles southern-hemisphere latitudes marked with a trailing S.

Why is 360 degrees in a full turn instead of 100?

Because of Babylonian sexagesimal counting, which used base 60 and produced a calendar and astronomy with 360 days in a year. The Babylonians divided the circle into 360 parts to match the year, and the convention stuck through Greek, Islamic and European astronomy into modern usage. The choice of 360 is convenient because it is divisible by so many small numbers, which made it easy to divide a circle into halves, thirds, quarters, sixths and so on without resorting to fractions.

Can I use this converter offline?

Yes. The conversion logic is a few lines of arithmetic and the page is a static bundle after the first load, so once the tool is cached by your browser it works without any further network requests. Bookmark it or save it as a desktop shortcut and it functions like a local utility.

Is the NATO mil the same as the mil used on rifle scopes?

Usually not. Most rifle scope reticles are graduated in true milliradians (1/(2000π) of a turn), because the manufacturer wants the convenient property that one mil of elevation moves the bullet one metre at a thousand metres for a fixed reference. Military and law-enforcement scopes that read in "mil" often mean NATO mils (1/6400 of a turn) instead, because the simpler integer divisions make manual ranging easier. Always check the reticle documentation before relying on the unit.

What is a gradian and who still uses it?

A gradian (also gon) divides the right angle into 100 equal parts, so a full turn is 400 gradians. It was introduced during the French Revolution as part of the metric system. Today it survives in some European surveying instruments, a few CNC controllers, and as the default angular unit in a small number of CAD packages. If you are converting data from an older theodolite, check whether the dial reads in gradians before assuming degrees.

References

• BIPM, The International System of Units (SI), 9th edition (2019). Bureau International des Poids et Mesures. Chapter 3 discusses the radian as the SI-coherent unit for angle.
• ISO 80000-3:2006, Quantities and units, Part 3: Space and time. International Organization for Standardization. Defines plane angle and solid angle, including the radian and steradian, and lists the conventional sexagesimal units (degree, arcminute, arcsecond).
• NIST, Units outside the SI. For the Angle Converter, National Institute of Standards and Technology. Summarises non-SI units accepted for use with the SI, including the degree, arcminute, arcsecond, and gradian, with their exact definitions.