Angle Converter
Unit ConversionConvert angles between degrees, radians, gradians, arcminutes, and arcseconds. Type any value into any field and every other unit updates instantly.
Angle
Other Units
1 Degree = 0.0174533 Radians ⟷ 1 Radian = 57.29578 Degrees
1 Degree = 1.111111 Gradians ⟷ 1 Gradian = 0.9 Degrees
1 Degree = 0.00277778 Turns (Revolutions) ⟷ 1 Turn (Revolution) = 360 Degrees
1 Degree = 17.453293 Milliradians ⟷ 1 Milliradian = 0.0572958 Degrees
1 Degree = 60 Arcminutes ⟷ 1 Arcminute = 0.0166667 Degrees
1 Degree = 3,600 Arcseconds ⟷ 1 Arcsecond = 0.000277778 Degrees
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Unit Conversion calculators
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SponsoredThis angle converter handles degrees, radians, gradians, turns, milliradians, arcminutes, and arcseconds at once: type a value into any box and every other unit updates immediately. It is the fastest way to move an angle between the units used in math, engineering, surveying, and astronomy.
Each unit is a fixed fraction of a full turn. A full circle is 360 degrees, 2π radians (about 6.2832), 400 gradians, 1 turn, about 6,283.2 milliradians, 21,600 arcminutes, or 1,296,000 arcseconds, so every conversion is exact.
Quick answer: 1 radian ≈ 57.2958°, 1 gradian = 0.9°, 1 turn = 360°, 1 milliradian ≈ 0.0573°, 1 arcminute = 1/60°, and 1 arcsecond = 1/3,600°. Negative angles convert the same way and describe a clockwise rotation.
For step by step formulas and tables for a single pair, use the dedicated converters linked below.
How to Convert Between Angle Units
Every conversion goes through degrees: radians = degrees × π/180, gradians = degrees × 10/9, turns = degrees ÷ 360, milliradians = degrees × 1,000π/180, arcminutes = degrees × 60, and arcseconds = degrees × 3,600. To go the other way, divide by the same factor.
Worked example: 45° = π/4 rad ≈ 0.785398 rad = 50 grad = 2,700′ = 162,000″.
Once an angle is in the unit you need, the Trigonometric Functions Calculator gives all six trig values in degrees or radians, and the Rotation Calculator rotates a point by an angle clockwise or counterclockwise.
Angle Units Compared
| Unit | Symbol | In Degrees | Full Circle |
|---|---|---|---|
| Degree | ° | 1 | 360 |
| Radian | rad | 57.29578 | 2π ≈ 6.2832 |
| Gradian (gon) | grad | 0.9 | 400 |
| Turn (revolution) | tr | 360 | 1 |
| Milliradian | mrad | 0.057296 | 2,000π ≈ 6,283.19 |
| Arcminute | ′ | 0.016667 | 21,600 |
| Arcsecond | ″ | 0.000278 | 1,296,000 |
Degrees are the everyday unit. Radians are the SI unit, the one calculus formulas use, and the one many software math functions expect. Gradians split a right angle into exactly 100 parts and appear in some surveying work. Turns (revolutions) count whole rotations and are the unit behind revolutions per minute. Milliradians are used for rifle scope adjustments and reticles, and arcminutes and arcseconds measure the small angles used in navigation, astronomy, and optics.
Types of Angles
Angles are named by their size. The same boundaries in every unit (acute, obtuse, and reflex angles fall strictly between the two values shown):
| Angle Type | Degrees | Radians | Turns | Gradians |
|---|---|---|---|---|
| Acute | 0° to 90° | 0 to π/2 | 0 to 1/4 | 0 to 100 |
| Right | 90° | π/2 | 1/4 | 100 |
| Obtuse | 90° to 180° | π/2 to π | 1/4 to 1/2 | 100 to 200 |
| Straight | 180° | π | 1/2 | 200 |
| Reflex | 180° to 360° | π to 2π | 1/2 to 1 | 200 to 400 |
| Full | 360° | 2π | 1 | 400 |
Two angles that add up to 90° are complementary, and two that add up to 180° are supplementary.
Angle Conversion Table
Common angles in every unit, plus one radian for reference:
| Degrees | Radians | Gradians | Turns | Arcminutes |
|---|---|---|---|---|
| 15° | 0.2618 rad | 16.6667 grad | 0.0417 tr | 900′ |
| 30° | 0.5236 rad | 33.3333 grad | 0.0833 tr | 1,800′ |
| 45° | 0.7854 rad | 50 grad | 0.125 tr | 2,700′ |
| 60° | 1.0472 rad | 66.6667 grad | 0.1667 tr | 3,600′ |
| 90° | 1.5708 rad | 100 grad | 0.25 tr | 5,400′ |
| 180° | 3.1416 rad | 200 grad | 0.5 tr | 10,800′ |
| 270° | 4.7124 rad | 300 grad | 0.75 tr | 16,200′ |
| 360° | 6.2832 rad | 400 grad | 1 tr | 21,600′ |
| 57.2958° | 1 rad | 63.662 grad | 0.1592 tr | 3,437.75′ |
Where Angle Conversion Is Used
Students convert between degrees and radians in trigonometry, spreadsheet and software users convert before calling trig functions that expect radians, surveyors meet gradians and arcseconds, and navigators and GPS users convert coordinates between decimal degrees and degrees, minutes, and seconds.
For angles outside 0° to 360°, the Coterminal Angle Calculator finds the equivalent angle, the Central Angle Calculator works out an angle from an arc and a radius, and the Scientific Calculator switches between degree and radian mode.
Frequently Asked Questions
How many degrees are in a radian?
One radian is 180/π degrees, about 57.2958°.
Why are there 360 degrees in a circle?
No one knows for certain. One theory links it to the roughly 360 days of a year, another to the base 60 counting of Babylonian astronomers. What is certain is that 360 is easy to divide: it has 24 divisors, so halves, thirds, quarters, sixths, and eighths of a turn are all whole degrees.
How do I convert degrees to radians in a spreadsheet?
In Excel, =RADIANS(A1) converts degrees to radians and =DEGREES(A1) converts radians back to degrees. Excel trig functions such as SIN take radians, so convert first.
How many arcminutes and arcseconds are in a degree?
One degree holds 60 arcminutes, and each arcminute holds 60 arcseconds, so a degree is 3,600 arcseconds.
How do I convert radians to revolutions?
Divide by 2π, about 6.2832, since one full revolution is 2π radians. For example, 10 rad ÷ 2π is about 1.5915 revolutions. Type the radians above and read the Turns box.
What is a milliradian?
A milliradian (mrad, often shortened to mil) is one thousandth of a radian, about 0.0573°. Rifle scope adjustments and reticles use it because 1 mrad covers about 1 unit at 1,000 units of distance, such as 10 cm at 100 m. Military maps and artillery sometimes use a different mil, 1/6,400 of a circle.
Can this converter handle negative angles?
Yes. A negative angle means a clockwise rotation, and every unit converts with the same sign.
How do I convert an angle to degrees, minutes, and seconds?
Use the Decimal Degrees to DMS Calculator, which splits a decimal angle such as 30.51° into 30° 30′ 36″ and can add N, S, E, or W for coordinates.
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