Angle Conversion Calculator
This calculator converts a single angle into all six common units at once: degrees, radians, gradians, turns, arc minutes and arc seconds. Choose your starting unit from the dropdown, whether degrees, radians, gradians (grads), turns, arc minutes or arc seconds, then type in the angle value, or click one of the common angle buttons (0, 30, 45, 60, 90, 180, 270 or 360 degrees) to fill it in automatically. As soon as you enter a number, the tool recalculates instantly and displays every equivalent value side by side in the Conversion Results panel and the Full Conversion Summary below it, so you can see, for example, that 90 degrees is the same as 1.5708 radians, 100 gradians, a quarter turn, 5,400 arc minutes and 324,000 arc seconds. A worked example beneath the results shows the exact formula used for each conversion, and a reference table lists the degree, radian, gradian, turn and arc second values for common angles such as 30, 45, 60, 90, 180 and 270 degrees. Use it to check trigonometry homework, convert between radians and degrees for programming and engineering calculations, work with gradians in surveying, or translate arc minutes and arc seconds from navigation and astronomy coordinates into degrees. Because these conversions are fixed mathematical relationships rather than measurements, the results are exact, not estimates.
1. Enter Angle
2. Conversion Results
Worked Example (default: 90 degrees)
Input: 90 degrees
Radians = 90 x (pi / 180) = pi / 2 = 1.570796...
Gradians = 90 x (400 / 360) = 90 x (10/9) = 100
Turns = 90 / 360 = 0.25 (a quarter circle)
Arc minutes = 90 x 60 = 5,400'
Arc seconds = 90 x 3,600 = 324,000''
Common Angle Reference Table
| Angle | Degrees | Radians | Gradians | Turns | Arc Seconds |
|---|---|---|---|---|---|
| Zero | 0 | 0 | 0 | 0 | 0 |
| 30 degrees (1/12 circle) | 30 | pi/6 ≈ 0.5236 | 33.33 | 0.0833 | 108,000 |
| 45 degrees (1/8 circle) | 45 | pi/4 ≈ 0.7854 | 50 | 0.125 | 162,000 |
| 60 degrees (1/6 circle) | 60 | pi/3 ≈ 1.0472 | 66.67 | 0.1667 | 216,000 |
| Right angle (1/4 circle) | 90 | pi/2 ≈ 1.5708 | 100 | 0.25 | 324,000 |
| Straight angle (1/2 circle) | 180 | pi ≈ 3.1416 | 200 | 0.5 | 648,000 |
| 3/4 circle | 270 | 3pi/2 ≈ 4.7124 | 300 | 0.75 | 972,000 |
| Full circle | 360 | 2pi ≈ 6.2832 | 400 | 1 | 1,296,000 |
Angle Units Explained
An angle describes the amount of rotation between two rays sharing a common endpoint. Several different units are used to measure angles, each suited to particular fields and applications.
Degrees
The degree is the most familiar unit of angle measurement. A full circle is divided into 360 degrees. This choice of 360 dates from ancient Babylon and was convenient because 360 has many divisors (it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180). Degrees are used in everyday navigation, geography, carpentry, and most applied contexts.
Radians
Radians are the standard unit of angle measurement in mathematics and physics. One radian is the angle subtended at the centre of a circle by an arc whose length equals the radius of the circle. A full circle is 2 x pi radians (approximately 6.2832). Radians are the natural unit for trigonometric functions and calculus because they simplify derivatives: d/dx(sin x) = cos x only holds when x is in radians. To convert degrees to radians: multiply by pi / 180. To convert radians to degrees: multiply by 180 / pi.
Gradians (Grads)
A gradian (also called grad or gon) divides the full circle into 400 equal parts. A right angle is exactly 100 gradians, a straight angle is 200, and a full circle is 400. Gradians were introduced during the French metric reform to facilitate decimal calculation. They are used in surveying, civil engineering, and some European technical fields. The conversion is: gradians = degrees x (400 / 360) = degrees x (10 / 9).
Turns
A turn (also called revolution or cycle) represents one complete rotation: 360 degrees, 2 x pi radians, or 400 gradians. Expressing angles as a fraction of a full turn is intuitive for rotational contexts such as gears, propellers, and clocks. For example, a quarter turn is 90 degrees and half a turn is 180 degrees.
Arc Minutes and Arc Seconds
Arc minutes and arc seconds subdivide the degree for high-precision angle measurements. One degree equals 60 arc minutes (symbol: '). One arc minute equals 60 arc seconds (symbol: ''). So one degree equals 3,600 arc seconds. These subdivisions are used in navigation, astronomy, cartography, and geographic coordinate systems. For example, a position given as 36 degrees 30 minutes south latitude equals 36.5 degrees south.
Conversion Formulas
| From | To | Formula |
|---|---|---|
| Degrees | Radians | rad = deg x (pi / 180) |
| Radians | Degrees | deg = rad x (180 / pi) |
| Degrees | Gradians | grad = deg x (10 / 9) |
| Gradians | Degrees | deg = grad x (9 / 10) |
| Degrees | Turns | turns = deg / 360 |
| Turns | Degrees | deg = turns x 360 |
| Degrees | Arc minutes | arcmin = deg x 60 |
| Degrees | Arc seconds | arcsec = deg x 3600 |
| Arc minutes | Degrees | deg = arcmin / 60 |
| Arc seconds | Degrees | deg = arcsec / 3600 |
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Sources and method: Conversion factors are exact mathematical relationships: 1 full circle = 360 degrees = 2 x pi radians = 400 gradians = 1 turn = 21,600 arc minutes = 1,296,000 arc seconds. These are definitional constants with no measurement uncertainty. Reference: ISO 80000-3 Quantities and units (Space and time).
This calculator performs exact mathematical conversions. Results beyond approximately 15 significant figures may show floating-point rounding inherent to all digital computation. For scientific work requiring exact rational expressions (such as pi/2 rather than 1.5708), note the exact form in the reference table above.