Angle Unit Converter

180° is π radians, and 1° is π/180 radians

A radian is the central angle that cuts off an arc as long as the circle's radius. A circle's circumference is 2π times its radius, so a full turn of 360° is 2π rad and a half turn of 180° is π rad. To convert degrees to radians, multiply by π/180. To convert radians to degrees, multiply by 180/π. One degree is about 0.017453 rad, and one radian is about 57.29578°.

Enter an angle and a unit in the converter above and the value in the target unit appears right away. When the target is radians, you get both the π fraction and the decimal, for example 45° = π/4 rad. In a radian field you can type 3π/4 or pi/6 and get degrees back. Nine units are shown in one list: degrees, radians, gradians, turns, arcminutes, arcseconds, degrees-minutes-seconds, milliradians and NATO mils. Dragging the protractor handle changes the value too. Negative angles and angles beyond 360° are also reduced to the equivalent angle within one turn.

How to use the converter

1. Type a number in the angle field and pick its unit. The default direction is degrees (°) to radians. Fractions such as 1/3 are fine, and a leading minus sign makes the angle negative.
2. When the unit is radians or milliradians, a π button appears next to the field. You can enter values with π, such as 3π/4, π/6 or 2π, and on a keyboard typing pi works as well.
3. When the unit is degrees, you can enter degrees, minutes and seconds as 37°33′59″ or 37 33 59. Values separated by spaces are read as degrees, minutes and seconds in that order.
4. Choose the target unit, or click a unit in the "The same angle in every unit" list. The ⇄ button in the middle swaps the two units and carries the current angle over as a value in the new unit.
5. Pick 2, 4, 6 or 10 decimal places. If a nonzero value is so small that it would show as 0, the tool shows three significant digits instead.
6. Drag the protractor handle, or select it and press the arrow keys to change the angle one step at a time. One step is 1° in degrees, π/180 in radians and 1 gon in gradians. The common-angle buttons (30° to 360°) enter the exact value in the current unit.

A radian counts how many radii fit along the arc

The BIPM, which maintains the International System of Units (SI), defines the radian as the angle subtended at the center of a circle by an arc equal in length to the radius, and notes that a right angle is π/2 rad. On a circle of radius 1, the arc length is the radian value itself. On a circle of radius r, an arc opening θ rad has length r × θ, and the sector area is r² × θ ÷ 2. That is why an angle written in degrees needs to be converted to radians before it goes into these formulas.

On the converter's protractor, the numbers 1, 2, 3… along the arc are the radian scale. The number goes up by one for every arc as long as the radius. A half turn holds a little over 3 of them (π ≈ 3.14) and a full turn a little over 6 (2π ≈ 6.28). The outer rim is labeled every 45° with both degrees and the π form.

How to calculate: multiply by π/180 and reduce

1. Degrees to radians: multiply the degrees by π/180. 60° × π/180 = 60π/180.
2. Write it as a π fraction: reduce the fraction of degrees over 180. 60/180 = 1/3, so 60° = π/3 rad. 135° gives 135/180 = 3/4, so 3π/4 rad, and 1° is π/180 rad.
3. Decimal radians: put 3.14159265… in for π. π/3 = 1.047198 rad.
4. Radians to degrees: multiply the radians by 180/π. 2 rad × 180/π = 114.591559°.
5. Radians with π to degrees: put 180° in place of π and no multiplication is needed. 5π/6 = 5 × 180° ÷ 6 = 150°.

When the angle in degrees is a whole number, or related to a divisor of 180 such as 30° or 45°, the π fraction comes out short. An angle with many decimals, such as 12.3456°, would need a huge denominator, so the converter shows a π fraction only when the denominator is 3,600 or less and shows the decimal alone otherwise.

Radians, gradians and turns for common angles

The nine units: definitions and where they are used

• Degrees, minutes, seconds: used in everyday life, maps and astronomy. The SI Brochure lists degree, minute and second among the units accepted for use with the SI.
• Radian: the SI unit of plane angle. Trigonometry, calculus, physics formulas and the math functions of programming languages take radians by default.
• Gradian (gon, grad): divides a right angle into 100, which makes calculations easy in surveying. It survives in some countries' surveying instruments and in the GRAD mode of scientific calculators.
• Turn (revolution): used where "how many turns" is the natural question, such as motor and wheel rotations or angular speed in rpm. One turn is 2π rad.
• Milliradian and NATO mil: used in rifle scopes, artillery firing data and military compasses. The names are similar but the sizes differ. The difference is explained in a section below.

Degrees-minutes-seconds to decimal degrees, and back

One degree is 60′ and one minute is 60″. To turn degrees, minutes and seconds into decimal degrees, compute degrees + minutes ÷ 60 + seconds ÷ 3,600. 37° 33′ 59.4″ is 37 + 33/60 + 59.4/3,600 = 37.5665°. In the other direction, multiply the fractional part by 60 twice. For 127.0276°, 0.0276 × 60 = 1.656 gives 1′, and the remaining 0.656 × 60 = 39.36 gives 39.36″, so the answer is 127° 1′ 39.36″.

The converter shows the seconds of a degrees-minutes-seconds result with four fewer decimals than the decimal setting. Since 1″ is about 0.000278°, six decimals leave the seconds with two decimal places. If rounding turns the seconds into 60″, it carries one minute so that a form like 59′ 60″ never appears. For converting many latitude and longitude coordinates with N/S and E/W marks, a dedicated latitude and longitude converter is more convenient.

Calculators, spreadsheets and code use radians by default

• Excel and Google Sheets: SIN, COS and TAN take radians. Convert a value in degrees with the RADIANS function. As in Microsoft's own example, =RADIANS(270) gives 4.712389 (3π/2), and DEGREES goes the other way. =SIN(RADIANS(30)) gives 0.5.
• JavaScript: Math.sin(30) is the sine of 30 radians, so it returns −0.988032, not 0.5. Write Math.sin(30 * Math.PI / 180) instead. Math.atan2 returns its result in radians from −π to π.
• Python: math.sin also takes radians. Convert with math.radians(30), and use math.degrees to see a resulting angle in degrees.
• Scientific calculators: the DEG, RAD or GRAD indicator on the display is the current mode. The same sin 30 gives 0.5 in DEG mode and −0.988032 in RAD mode, so check the mode before calculating.

Negative angles and angles beyond 360° within one turn

Adding or subtracting 360° does not change the direction an angle points. So 450°, 90° and −270° all point the same way. The converter converts the angle you enter as it is, and also shows the same direction reduced to the 0°–360° range and to the −180°–180° range. Counterclockwise is positive, and 0° points to the right (the +x axis).

If you do the same calculation in code, note that the % operator in JavaScript returns a negative remainder for negative numbers (−90 % 360 = −90). To get the 0°–360° range, use ((x % 360) + 360) % 360, which adds and takes the remainder once more.

A milliradian and a mil are different units

A milliradian (mrad) is 1/1,000 rad, so a full turn is 2,000π ≈ 6,283.19 mrad. At 1,000 m, 1 mrad covers a width of about 1 m, which makes it handy for estimating distance and size through a scope. A NATO mil, on the other hand, divides a full turn into 6,400, and the NIST conversion table lists 1 mil = 0.05625°. So 1 mrad (about 0.057296°) is roughly 1.9% larger than 1 NATO mil.

The size of a mil also varies by country and era. The former Soviet Union and Finland used 6,000 per turn, and Sweden used 6,300 until 2007. The mil in this converter is the NATO mil (6,400). If equipment or a map says only "mil", check the manual to see how many make a full turn.

Common mistakes

• Wrong calculator mode: entering an angle in degrees in RAD mode, or a radian value in DEG mode, gives a completely different answer. Excel and programming languages default to radians.
• Rounding π to 3.14: computing 180° as 3.14 rad leaves you 0.0016 rad (about 0.09°) short of the real π rad. The error adds up for large angles or repeated multiplication, so use a π fraction or enough digits.
• Writing 37°33′ as 37.33°: minutes are base 60, so 37°33′ is 37.55°. Do not mix decimal degrees with degree-minute notation.
• Using quote marks for minutes and seconds: ′ (prime) and ″ (double prime) are the proper symbols. The converter also reads ' and " typed on a keyboard as minutes and seconds, but when you move the text into a document, check that they are not confused with quotation marks.
• Mixing compass bearings with math angles: map and navigation bearings start at 0° at north and go clockwise. Math angles start at 0° on the right and go counterclockwise. To convert, compute bearing = 90° − math angle, then reduce it to within one turn. A math angle of 30° is a bearing of 60°.
• Treating gradians and percent grade as the same: 100 gon is a right angle (90°), but a 100% grade is a slope where the rise equals the run, which is 45°. Percent grade is not an angle unit.

Limits and cautions

• This tool converts plane angles only. It does not convert solid angles (steradians), percent grade or bearing systems.
• Results that involve π (degrees to radians and back) are endless decimals, so they are approximations rounded to the chosen number of places. Those are marked with "≈". When a π fraction can express the value exactly, the tool shows "=" together with the π fraction.
• Only one mil is supported: the NATO mil (1/6,400 of a turn). For mils that divide the circle into 6,000 or 6,300, multiply the turn value by that number.
• Angles whose absolute value exceeds one billion degrees are not calculated.

Basis: unit definitions follow the BIPM SI Brochure (9th edition) and NIST SP 811 Appendix B.9 (updated August 18, 2025), and they do not change from year to year. The function descriptions from Microsoft, MDN and the Python documentation were checked on October 20, 2026.

Sources

• 01BIPM — The International System of Units (SI Brochure, 9th edition)Definition of the radian (an arc equal in length to the radius; a right angle is π/2 rad), Table 8: 1° = π/180 rad, 1′ = 1/60°, 1″ = 1/60′
• 02NIST Guide to the SI — Appendix B.9gon = 0.9°, mil = 0.05625°, revolution = 6.283185 rad
• 03Microsoft Support — RADIANS functionConverts degrees to radians; =RADIANS(270) = 4.712389 (3π/2)
• 04MDN — Math.atan2()The result is in radians from −π to π
• 05Python documentation — math modulemath.radians and math.degrees; math.sin takes radians
• 06Wikipedia — MilliradianNATO 6,400; former USSR and Finland 6,000; Sweden 6,300 (before 2007) mils; 1 mrad ≈ 1 m at 1,000 m

Frequently asked questions

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