📐 Angle Converter: Degrees, Radians, Gradians and More

What is the Angle Converter?

An angle measures the amount of turn between two rays that share a common endpoint, called the vertex. The same turn can be written in several units, which is why a single full rotation reads as 360 degrees, 2 pi radians, 400 gradians or 1 revolution depending on the system you pick. Each unit is simply a different sized slice of the same circle, so converting between any two of them is a matter of scaling by one fixed factor. This tool pivots every conversion through degrees, so any pair of units always lines up correctly.

The two units you meet most often are degrees and radians. Degrees split a full circle into 360 equal parts, a convention inherited from ancient Babylonian astronomy because 360 divides cleanly by so many whole numbers. Radians instead measure an angle by the arc length it sweeps on a unit circle, so a full circle is 2 pi radians and a half circle is pi radians. Radians are the natural unit in calculus and physics because the formulas for derivatives, Taylor series and rotational motion only come out clean when angles are measured in radians. To convert, multiply degrees by pi and divide by 180; to go back, multiply radians by 180 and divide by pi.

Gradians, also called gons or grades, divide a circle into 400 parts so that a right angle is exactly 100 gradians. This decimal friendly layout is convenient for surveying and some European civil engineering, where slopes and bearings are easier to add when a quarter turn is a round 100. One degree equals 10 divided by 9, or about 1.1111 gradians.

Arcminutes and arcseconds split a single degree into smaller pieces: one degree is 60 arcminutes and one arcminute is 60 arcseconds, so a degree holds 3600 arcseconds. These tiny units are the standard for astronomy, optics, telescope resolution and precise navigation, where the angle between two stars or the wobble of a distant planet is far smaller than a single degree. The same 60 and 3600 split underlies degrees-minutes-seconds (DMS) coordinates used in latitude and longitude.

Milliradians and mils handle the small angles common in ballistics and ranging. A true milliradian is one thousandth of a radian, so a full circle holds about 6283.19 milliradians. The military NATO mil is rounded to exactly 6400 per circle for easy artillery math, which is close to but not identical to a true milliradian. At long range one mil subtends roughly one meter at one kilometer, which is what makes mil based scopes useful for estimating distance to a target of known size.

Because all of these units describe the same physical turn, the only thing that ever changes is the size of the counting unit. Once you know how many of each unit fit in a full circle, every conversion is just a ratio. This converter stores each value internally at full precision and rounds only the number it shows you, so chained conversions do not drift the way they would if you rounded pi or a factor too early by hand.

When to use it

• Converting a degree value into radians before calling a trig function in JavaScript, Python, C or a spreadsheet, since almost all programming languages expect radian input.
• Turning the radian output of a physics, calculus or engineering problem back into degrees that are easier to picture and sketch.
• Reading a surveying or civil engineering angle given in gradians and expressing it in degrees for a drawing or CAD file.
• Breaking a small astronomical or optical angle into arcminutes and arcseconds, or reading a DMS latitude and longitude as decimal degrees.
• Converting milliradians or NATO mils for rifle scope adjustments, artillery aiming and target range estimation.
• Checking homework or exam answers where the same angle must be expressed in two or more units.

How to use the Angle Converter

1. Type the angle you have into the box for its unit (for example, enter 90 in the Degrees box).
2. Read the equivalent value from every other unit box, which updates live as you type.
3. Use the 90, 180 or 360 preset buttons to load a common degree value quickly.
4. Copy the result you need straight from its box into your code, spreadsheet or drawing.
5. Press Clear to empty all boxes and start a fresh conversion.

Formula & method

Worked examples

Common angles across units

How each unit divides a full circle

Conversion factors from degrees

Common mistakes to avoid

• Feeding degrees into a function that expects radians. Trig functions in most programming languages and spreadsheets take radians, not degrees. Passing 90 instead of pi ÷ 2 (about 1.5708) gives wildly wrong results. Convert the angle first, or use a dedicated degrees mode if your tool offers one.
• Confusing gradians with degrees. A right angle is 90 degrees but 100 gradians. Because the numbers look similar near small angles, it is easy to mix them up. Always check which unit your instrument, scope or software output is actually reporting before you convert.
• Mixing up arcminutes and minutes of time. The arcminute symbol (a single prime) also appears for minutes of time and for feet. An arcminute is one sixtieth of a degree, not a unit of time. Read the context before converting so you do not divide by the wrong 60.
• Adding DMS values as if they were decimal. Degrees, arcminutes and arcseconds use base 60, not base 10. You cannot simply write 30 degrees 30 arcminutes as 30.30 degrees. The correct decimal value is 30 + 30 ÷ 60 = 30.5 degrees. Convert to decimal first, then add.
• Treating a NATO mil as a true milliradian. A true milliradian gives about 6283.19 per circle, but the NATO mil is rounded to exactly 6400 per circle. They are close but not equal, so using one in place of the other introduces a small but real aiming error at long range.
• Rounding pi too early. Using 3.14 instead of a fuller value of pi introduces a visible error in radian conversions, and the error compounds across chained steps. This tool keeps full precision internally and rounds only the displayed result, so do the same in your own calculations.

Glossary

Degree

An angle unit equal to one 360th of a full circle. Written with the degree symbol after the number.

Radian

The angle that sweeps an arc equal to the radius on a circle. A full circle is 2 pi radians. The SI unit for angle.

Gradian (gon)

An angle unit equal to one 400th of a full circle, used in surveying. A right angle is exactly 100 gradians.

Arcminute

One sixtieth of a degree, used for small angles in astronomy, optics and navigation. Marked with a single prime.

Arcsecond

One sixtieth of an arcminute, or one 3600th of a degree, used for very small angles. Marked with a double prime.

DMS

Degrees, minutes and seconds notation, where a degree is split into 60 arcminutes and each arcminute into 60 arcseconds. Common for map coordinates.

Milliradian (mrad)

One thousandth of a radian. A full circle holds about 6283.19 milliradians. Used in ballistics and ranging.

Mil (NATO)

A military angle unit rounded to exactly 6400 per circle for artillery and scope adjustments, close to a true milliradian.

Revolution (turn)

One full rotation, equal to 360 degrees or 2 pi radians. Also called a turn or a cycle.

Frequently asked questions