Angle unit conversion table
Each unit as a fraction of a full turn, with its value in degrees and radians.
| Unit | Degrees | Radians | Fraction of a turn |
|---|---|---|---|
| 1 turn | 360 | 6.2831853 | 1 |
| 1 degree | 1 | 0.017453293 | 1/360 |
| 1 gradian | 0.9 | 0.015707963 | 1/400 |
| 1 radian | 57.295780 | 1 | 1/(2π) |
| 1 arcminute | 0.016666667 | 0.00029088821 | 1/21,600 |
| 1 arcsecond | 0.00027777778 | 0.0000048481368 | 1/1,296,000 |
| 1 milliradian | 0.057295780 | 0.001 | 1/(2000π) |
- Every factor in this table is exact by definition. The decimal values shown are rounded, but the underlying relationships — 360 degrees, 400 gradians and 2π radians to a turn — carry no measurement uncertainty at all.
- The milliradian used here is the true angular milliradian, 1/1000 radian. Several shooting disciplines use a slightly different 'mil' — the NATO mil divides the turn into 6,400 parts rather than 6,283.19 — so check which one a scope or reticle means before converting.
- Spreadsheet and programming trigonometric functions almost always expect radians. Passing degrees to them is one of the most common sources of silently wrong geometry.
What are the common angle units?
An angle measures rotation, and every unit is a fraction of a full turn. The degree divides the turn into 360 parts, a convention inherited from Babylonian astronomy that survives because 360 divides evenly by so many numbers. The radian divides it into 2π parts and is the SI unit: one radian is the angle subtending an arc equal in length to the radius.
The radian is not an arbitrary choice. Calculus assumes it — the derivative of sin x equals cos x only when x is in radians — and so do the arc-length and sector-area formulas, which is why an angle entered in degrees must be converted before any of them apply.
The gradian, also called the gon, divides the turn into 400 parts so that a right angle is exactly 100 gon. It is used in surveying, particularly in continental Europe. Arcminutes and arcseconds subdivide the degree by 60 and 3,600 for astronomy and navigation, and the milliradian is used in optics and ballistics, where one milliradian subtends approximately one metre at one kilometre.
How to use this angle converter
- Enter the angle you want to convert. Negative values are accepted — a bearing or a rotation can run either way.
- Select its unit: degrees, radians, gradians, turns, arcminutes, arcseconds or milliradians.
- Read the equivalent in every other unit at once.
- Use radians for anything entering a trigonometric formula, a derivative or an arc-length calculation.
The conversion factors behind the angle converter
The value is converted to radians, then out to each target unit. Because a full turn is 2π radians and also 360 degrees, 400 gradians and 1 turn, every factor follows from those identities exactly.
Worked example: 45° in radians is 45 × π/180 = π/4 = 0.78539816 rad. In gradians: 45 × 400/360 = 50 gon exactly. In turns: 45/360 = 0.125 exactly. In arcseconds: 45 × 3,600 = 162,000 exactly.
Common mistakes
- Passing degrees to a trigonometric function that expects radians. sin(30) is −0.988 if 30 is read as radians, not the 0.5 you wanted.
- Confusing the angular milliradian (1/1000 rad, 6,283.19 per turn) with the NATO mil (6,400 per turn). They differ by about 1.9%.
- Reading arcminutes as minutes of time. An arcminute is 1/60 of a degree; the symbols ′ and ″ mean angle here, not duration.
- Assuming a right angle is 90 in every unit. It is 90°, but 100 gon, π/2 rad and 0.25 turn.
अक्सर पूछे जाने वाले सवाल
How many degrees is 1 radian?
57.295780 degrees, or 180/π. The relationship follows from a full turn being both 360 degrees and 2π radians, so one radian is 360/(2π) = 180/π degrees. The reverse conversion is 1° = π/180 ≈ 0.017453293 rad.
Why do mathematicians use radians instead of degrees?
Because the calculus only works cleanly in radians. The derivative of sin x is cos x, and the small-angle approximation sin x ≈ x, both hold only when x is measured in radians. Arc length (rθ) and sector area (½r²θ) likewise assume radians. In degrees each of these picks up a factor of π/180.
What is a gradian used for?
Surveying, mainly in continental Europe. Dividing the turn into 400 parts makes a right angle exactly 100 gon, so quadrants and slopes fall on round numbers — a convenience when reading a theodolite. It is also called the gon or grade.
How many arcseconds are in a degree?
3,600. A degree divides into 60 arcminutes, and each arcminute into 60 arcseconds, giving 3,600 arcseconds per degree and 1,296,000 in a full turn. Astronomers use these subdivisions because celestial positions demand far finer resolution than a degree offers.
What is a milliradian in practical terms?
One thousandth of a radian, which subtends almost exactly one metre at one kilometre — the property that makes it useful in optics and ranging. Note that some shooting disciplines use a 'mil' defined as 1/6400 of a turn instead, which is about 1.9% different.
संदर्भ
- Bureau International des Poids et Mesures (BIPM). The International System of Units (SI Brochure), 9th edition, 2019 — the radian as the coherent SI unit of plane angle, and the degree, minute and second accepted for use with the SI.
- NIST Special Publication 811 (Thompson & Taylor). Guide for the Use of the International System of Units (SI), 2008 — plane-angle conversion factors.