Unit & Measurement Tables
Angle Conversion Table
The radian is the only angle unit that is not arbitrary. It is defined as the arc length divided by the radius, which makes it dimensionless — a ratio of two lengths. Every other unit is a scale factor on top of it: a degree is π/180 radians because someone in Babylon decided a full turn should be 360 parts, and a gradian is π/200 because French surveyors preferred counting in hundreds.
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That history explains why the conversions look awkward. There is no exact decimal for 1 radian in degrees; it is 57.295 779 513 082 32 and continues forever. Anything that claims otherwise is rounding. The tables here compute from 180/π at build time and print six decimals, which is enough for surveying and far more than most work needs.
Two practical notes. First, spreadsheets and programming languages almost always take trigonometric functions in radians, so a formula written in degrees has to be converted first — this is the single most common source of wrong answers in engineering homework. Second, small angles get their own units because degrees are too coarse: an arcsecond is 1/3,600 of a degree, and astronomy, optics and ballistics all work in arcseconds or milliradians rather than degrees.
Degrees to radians, gradians, turns, arcminutes and arcseconds
| Degrees | Radians | Gradians | Turns | Arcminutes | Arcseconds |
|---|---|---|---|---|---|
| 0.5 | 0.008727 | 0.56 | 0.0014 | 30 | 1,800 |
| 1 | 0.017453 | 1.11 | 0.0028 | 60 | 3,600 |
| 5 | 0.087266 | 5.56 | 0.0139 | 300 | 18,000 |
| 15 | 0.261799 | 16.67 | 0.0417 | 900 | 54,000 |
| 30 | 0.523599 | 33.33 | 0.0833 | 1,800 | 108,000 |
| 45 | 0.785398 | 50.00 | 0.1250 | 2,700 | 162,000 |
| 60 | 1.047198 | 66.67 | 0.1667 | 3,600 | 216,000 |
| 90 | 1.570796 | 100.00 | 0.2500 | 5,400 | 324,000 |
| 120 | 2.094395 | 133.33 | 0.3333 | 7,200 | 432,000 |
| 180 | 3.141593 | 200.00 | 0.5000 | 10,800 | 648,000 |
| 270 | 4.712389 | 300.00 | 0.7500 | 16,200 | 972,000 |
| 360 | 6.283185 | 400.00 | 1.0000 | 21,600 | 1,296,000 |
A right angle is 90 degrees, 100 gradians, a quarter turn, or 1.570 796 radians. Only the radian is an SI unit.
Radians to degrees, gradians and turns
| Radians | Degrees | Gradians | Turns |
|---|---|---|---|
| 0.100000 | 5.730 | 6.37 | 0.0159 |
| 0.250000 | 14.324 | 15.92 | 0.0398 |
| 0.500000 | 28.648 | 31.83 | 0.0796 |
| 1.000000 | 57.296 | 63.66 | 0.1592 |
| 1.570796 | 90.000 | 100.00 | 0.2500 |
| 2.000000 | 114.592 | 127.32 | 0.3183 |
| 3.141593 | 180.000 | 200.00 | 0.5000 |
| 4.712389 | 270.000 | 300.00 | 0.7500 |
| 6.283185 | 360.000 | 400.00 | 1.0000 |
π radians is exactly 180 degrees, which is why radians are the natural unit for calculus — the derivative of sine is cosine only in radians.
Small angles in arcminutes, arcseconds and milliradians
| Degrees | Arcminutes | Arcseconds | Milliradians |
|---|---|---|---|
| 0.001 | 0.06 | 3.6 | 0.0175 |
| 0.010 | 0.60 | 36.0 | 0.1745 |
| 0.050 | 3.00 | 180.0 | 0.8727 |
| 0.100 | 6.00 | 360.0 | 1.7453 |
| 0.500 | 30.00 | 1,800.0 | 8.7266 |
| 1.000 | 60.00 | 3,600.0 | 17.4533 |
| 2.000 | 120.00 | 7,200.0 | 34.9066 |
| 5.000 | 300.00 | 18,000.0 | 87.2665 |
One milliradian is 3.438 arcminutes, or roughly 1 metre at 1 kilometre. Telescopes and rifle scopes both work in these units.
Angles you meet in ordinary geometry
| Angle | Degrees | Radians | Gradians |
|---|---|---|---|
| Minute hand, one minute | 6 | 0.104720 | 6.67 |
| Earth rotation, one hour | 15 | 0.261799 | 16.67 |
| Equilateral triangle corner | 60 | 1.047198 | 66.67 |
| Square corner | 90 | 1.570796 | 100.00 |
| Regular pentagon interior | 108 | 1.884956 | 120.00 |
| Regular hexagon interior | 120 | 2.094395 | 133.33 |
| Straight line | 180 | 3.141593 | 200.00 |
| Full turn | 360 | 6.283185 | 400.00 |
The interior angle of a regular n-gon is (n − 2) × 180 / n degrees. The pentagon and hexagon rows follow that rule.
How to Use These Tables
- Feeding a spreadsheet: multiply degrees by 0.017453 to get radians before calling SIN, COS or TAN. A 30 degree slope is 0.5236 radians.
- Surveying: multiply gradians by 0.9 to get degrees. A 400 gradian circle is 360 degrees, so the units line up exactly at the full turn.
- Optics and shooting: one milliradian is about 1 metre at 1,000 metres, or 3.438 arcminutes. A scope that adjusts in 0.1 mil moves the point of impact 1 cm at 100 m.
- Astronomy: multiply arcseconds by 0.0002778 to get degrees. The Moon is about 1,860 arcseconds across, which is why it looks 30 arcminutes wide.
- Geometry: interior angle of a regular n-gon is (n − 2) × 180 / n degrees. Convert the result to radians by dividing by 57.2958.
Common Questions
How many degrees is one radian?
57.295 779 513 082 32 degrees, and the digits never terminate. It is 180 divided by π, so any decimal you write is a rounded value.
Why do calculators have a degree and a radian mode?
Because the trigonometric functions only behave correctly in radians. The derivative of sine is cosine only when the input is in radians; in degree mode the calculator silently applies a scale factor so the answers look right.
What is a gradian used for?
Land surveying in continental Europe. A right angle is 100 gradians, so bearings come out as clean two-digit numbers instead of 90, 45 and 22.5. It is also called a gon.
How many arcseconds are in a degree?
3,600. One degree is 60 arcminutes and one arcminute is 60 arcseconds, following the same base-60 pattern as hours, minutes and seconds.
What is a milliradian in practical terms?
About 1 metre at 1 kilometre, because a milliradian is 0.001 radians and the small-angle approximation makes the arc length equal the offset. It is 3.438 arcminutes, or 0.0573 degrees.
Is a turn an official unit?
Not in the SI, but it is unambiguous and common in engineering: one turn is 2π radians, or 360 degrees. Motor datasheets often quote steps per turn rather than radians.
More Unit & Measurement Tables
Sources
- BIPM — The International System of Units (SI Brochure), 9th edition — Bureau International des Poids et Mesures
- NIST Special Publication 811 — units for plane angle — National Institute of Standards and Technology
- ISO 80000-3 — quantities and units, space and time — International Organization for Standardization
Filed with Unit Conversion calculations.