How to Convert Degrees to Radians (and Other Angle Units)

Most of us learn angles in degrees: 90° for a corner, 360° for a full turn. Then a calculator returns nonsense for sin(90), a programming function wants radians, or a map gives a coordinate as 40° 42′ 46″. This guide explains where each angle unit comes from, how to convert between them and the mistakes that trip people up most often.

Why there are so many angle units

• Degrees split a full turn into 360 parts, a number that divides evenly by 2, 3, 4, 5, 6, 8, 9, 10 and 12. That keeps everyday fractions of a circle tidy.
• Radians measure an angle by the arc it cuts: one radian is the angle at which the arc is as long as the radius. A full turn is 2π radians, about 6.283.
• Gradians split a turn into 400 parts, so a right angle is exactly 100. They appear in some surveying work.
• Arcminutes and arcseconds divide a degree into 60 and then 3,600 parts, for very fine angles in astronomy, navigation and maps.
• Mils are used for aiming and estimating distance. The NATO mil is 1/6,400 of a turn; the milliradian is 1/1,000 of a radian.

Why mathematics prefers radians

Radians are not just another unit; they make formulas simpler. With an angle θ in radians:

• The arc length on a circle of radius r is simply s = r × θ.
• The area of a slice of a circle is ½ × r² × θ.
• For small angles, sin θ ≈ θ. At 0.1 rad (about 5.7°), sin 0.1 = 0.0998.

In degrees, every one of these needs an extra factor of π/180. That is why calculus, physics and nearly all programming languages use radians by default.

The conversion formulas

Everything rests on one fact: 180° = π radians.

• Degrees to radians: multiply by π/180.
• Radians to degrees: multiply by 180/π.
• Degrees to gradians: divide by 0.9.
• Degrees to NATO mils: multiply by 6,400/360, about 17.78.

Common angles worth memorizing

• 30° = π/6 ≈ 0.5236 rad
• 45° = π/4 ≈ 0.7854 rad
• 60° = π/3 ≈ 1.0472 rad
• 90° = π/2 ≈ 1.5708 rad
• 180° = π ≈ 3.1416 rad
• 270° = 3π/2 ≈ 4.7124 rad
• 360° = 2π ≈ 6.2832 rad

Worked examples

2.5 radians to degrees. 2.5 × 180/π ≈ 143.2394°. To write it in degrees, minutes and seconds, take the whole degrees (143), multiply the remaining 0.2394 by 60 to get 14.367 minutes, then multiply the remaining 0.367 by 60 for about 22 seconds. The precise result is 143° 14′ 22.016″.

A map coordinate to decimal degrees. Sydney lies at roughly 33° 51′ 54.5″ S. In decimal: 33 + 51/60 + 54.5/3,600 = 33.865139. South latitudes are negative, so the value to type into a mapping app is −33.865139.

Decimal degrees back to DMS. New York's 40.7128° N, 74.0060° W becomes 40° 42′ 46.08″ N and 74° 0′ 21.6″ W.

How precise is a coordinate?

Angle units become very concrete on a map. One degree of latitude is about 111 km, so:

• 1 arcminute of latitude ≈ 1.85 km, which is where the nautical mile comes from.
• 1 arcsecond ≈ 31 m.
• 0.0001° ≈ 11 m, and 0.000001° ≈ 11 cm.

A coordinate given to six decimal places pins a point down to roughly the width of a hand, which is already finer than a phone's GPS can deliver. Degrees of longitude shrink as you move away from the equator, so these distances apply to latitude everywhere and to longitude only near the equator.

Mils and milliradians at a distance

The appeal of milliradians is that they scale neatly with distance: an object 1 m wide seen from 1,000 m away covers about 1 mrad. One NATO mil is about 0.9817 mrad, so at the same distance it spans about 0.98 m. The two are close but not equal, and the gap matters over long ranges. Check which one a scope, compass or manual uses before converting.

How to use the Angle Converter

Type an angle, choose the unit you have and the unit you want, and the Angle Converter shows the result along with a table of the same angle in all eight units: degrees, radians, gradians, turns, arcminutes, arcseconds, milliradians and NATO mils. It also writes the angle in degrees, minutes and seconds. A second panel turns degrees, minutes and seconds, such as a map coordinate, into decimal degrees, with a sign for south and west.

Common mistakes

• Calculator in the wrong mode. If sin(90) gives 0.894 instead of 1, the calculator is in radian mode and has used 90 radians. Look for a DEG or RAD indicator.
• Forgetting that code expects radians. Math.sin(30) in JavaScript returns −0.988, not 0.5. Convert first: Math.sin(30 * Math.PI / 180).
• Reading minutes as decimals. 40° 30′ is 40.5°, not 40.30°. Minutes are sixtieths of a degree.
• Losing the sign. Enter 74° W as a positive longitude and New York moves to Central Asia.
• Rounding π too early. Using 3.14 turns 90° into 1.5700 rad instead of 1.5708. Keep full precision until the last step.

FAQ

Why is a circle 360 degrees?

The choice is usually traced back to ancient Babylonian astronomy, which counted in base 60, and to the year having roughly 360 days. The number stuck because it divides so evenly.

What is a steradian?

It is the three-dimensional cousin of the radian and measures a solid angle, such as the cone of light from a lamp. A full sphere is 4π steradians. It is not an angle on a flat plane, so it cannot be converted to degrees.

Is a radian a real unit?

Strictly, a radian is a ratio of two lengths, arc over radius, so it has no dimension. That is why formulas often drop the "rad" label, and a plain number such as 1.5708 is understood to mean radians.

How many degrees is one radian?

180/π, which is about 57.29578°, or 57° 17′ 44.8″.