Angle Converter

How do you convert angle units?

To convert any angle, express it in degrees, then divide by the size of the target unit in degrees. The key link is π rad = 180°, so 1 radian is 180/π ≈ 57.29578°, and 1° is π/180 ≈ 0.0174533 rad.

Degrees and radians

• Degrees to radians: multiply by π/180.
• Radians to degrees: multiply by 180/π.

Worked example: 35° × π/180 = 0.610865 rad. In reverse, 2.5 rad × 180/π = 143.2394°. The degrees to radians converter and the radians to degrees converter do this instantly for any value.

What a radian actually is

A radian is defined by arc length: an angle in radians is the length of the arc it cuts from a circle divided by the circle’s radius. When the arc is exactly as long as the radius, the angle is 1 radian. A full circle has a circumference of 2πr, so a full turn is 2π radians.

This is why radians dominate mathematics and physics. Formulas such as arc length s = rθ, angular speed and the derivatives of sine and cosine only take their simple form when θ is in radians. Most programming languages also expect radians: JavaScript’s Math.sin and Python’s math.sin both take radians, so a value in degrees must be converted first.

Degrees, minutes and seconds

For fine angles, a degree splits into 60 arcminutes (′), and each arcminute into 60 arcseconds (″). To turn a decimal angle into this format, multiply the decimal part by 60 for minutes, then the remaining decimal by 60 for seconds. A latitude of 51.4778° becomes 51° 28.668′, or 51° 28′ 40.08″. Pick arcminutes in the converter at the top of this page to handle the first step.

What is the SI unit of angle, and how is it defined?

The radian (rad) is the SI unit of plane angle. The BIPM SI Brochure (9th edition, 2019) lists it as a derived unit with the special name radian and defines it as 1 rad = 1 m/m: the length of an arc divided by the radius of its circle. Because it is a ratio of two lengths, the radian is dimensionless, which is why an angle in radians can go straight into formulas such as arc length s = rθ with no conversion constant. The same brochure defines the steradian (sr), the unit of solid angle, as 1 m²/m².

The degree (°), arcminute (′) and arcsecond (″) are not SI units, but the SI Brochure lists them among the non-SI units accepted for use with the SI and defines the degree as exactly π/180 rad, about 0.0174533 rad. ISO 80000-3, the international standard for quantities of space and time, uses the same definitions. Going the other way, one radian is about 57.29578°, as the radians to degrees converter shows.

Why a circle has 360 degrees

The division of a degree into 60 minutes and each minute into 60 seconds comes from Babylonian astronomy, which counted in base 60 (the sexagesimal system). The number 360 has lasted because it divides evenly by 24 whole numbers, including 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20 and 24, so halves, thirds, quarters, fifths, sixths and eighths of a turn are all whole degrees. A full turn is 360°, 2π rad (about 6.283185 rad), 400 gon or exactly 1 revolution.

Which angle units are used where?

Degrees are the everyday unit, radians are the standard in mathematics, physics and programming, and gradians survive mainly in surveying; arcminutes, arcseconds and revolutions each serve a specialist field.

• Degrees: everyday angles everywhere, from geometry homework and compass bearings to roof pitches, saw settings and map coordinates. For slopes on building projects, see the construction calculators.
• Radians: mathematics, physics and engineering, as well as programming, graphics and game engines, where trigonometric functions expect radians.
• Gradians (gon): surveying and civil engineering, especially in continental European countries such as Germany, Switzerland and France, where a right angle of exactly 100 gon simplifies calculations. Many scientific calculators include a GRAD mode. The degrees to gradians converter translates survey data in either direction.
• Arcminutes and arcseconds: navigation, where latitude and longitude are written in degrees and minutes, and astronomy, where star positions, planet sizes and telescope resolution are given in arcseconds. Rifle scopes use minutes of angle (MOA): 1 MOA spans about 1.047 inches at 100 yards.
• Revolutions (turns): mechanical engineering, rotary encoders, motors and gears. Rotational speed in revolutions per minute is covered by the frequency converter.

The nautical mile has an angle behind it: it was originally based on one arcminute of latitude, which is why it is used in sea and air navigation. It is now defined as exactly 1,852 m. See the length converter for nautical miles.

What is the difference between MOA and a milliradian (mil)?

One minute of angle (MOA) is 1/60 of a degree, while one milliradian (mrad, or mil) is 1/1,000 of a radian, so 1 mil ≈ 3.4377 MOA. Both are tiny angles used to adjust rifle scopes, read reticles and lay artillery, and mixing them up moves the point of aim by more than a factor of three.

• MOA: 1 MOA spans about 1.0472 in at 100 yards, or 2.9089 cm at 100 m. Many shooters round it to 1 inch at 100 yards, a convenience sometimes called shooter’s MOA. Scope turrets often click in ¼ MOA steps, about 0.2618 in at 100 yards.
• Milliradian: 1 mrad spans 10 cm at 100 m and about 3.6 in at 100 yards, so a 0.1 mil click moves the impact about 1 cm at 100 m. The metric scaling also makes range estimation simple: distance in meters = target size in meters × 1,000 ÷ size in mils, so a 1.8 m target spanning 2 mils is about 900 m away.
• Military mils: a full turn holds 2π × 1,000 ≈ 6,283.19 true milliradians. NATO rounds this to 6,400 mils per circle, so one NATO mil is 0.05625°, slightly smaller than a true milliradian.

To convert between the two, go through degrees: MOA ÷ 60 gives degrees, and degrees × π/180 × 1,000 gives milliradians, so 1 MOA ≈ 0.2909 mrad. The degrees to radians converter covers the middle step.