Use this Angle Unit Converter to enter values, adjust options, and review results in a compact responsive workspace.
Results are calculated automatically as you enter data.
Enter an angle and choose its measure to convert across common angle units.
▼ See explanations and tips below ▼
Angle unit conversion means rewriting the same amount of rotation in a different unit. A quarter turn, for example, can be written as \(90^\circ\), \(\frac{\pi}{2}\) radians, \(100\) gradians, \(0.25\) turn, \(5{,}400\) arcminutes, or \(324{,}000\) arcseconds. The angle has not changed; only the measurement scale has changed.
This matters because different fields prefer different angle units. Degrees are familiar in geometry and everyday measurement. Radians are common in trigonometry, calculus, and physics because they connect angles directly to arc length. Arcminutes and arcseconds are useful for very small angles, especially in astronomy, mapping, surveying, and precision measurement. Gradians, also called gons, divide a right angle into \(100\) parts and may appear in surveying or GIS contexts. Turns describe rotation as a fraction or multiple of a full revolution.
The key idea is that all these units are tied to one full rotation. Once you know the value of a full rotation in each unit, converting between units becomes a matter of multiplying by a fixed conversion factor.
Using the wrong angle unit can change the meaning of a calculation completely. An angle of \(2\) radians is about \(114.59^\circ\), not \(2^\circ\). A calculator, graphing tool, or trigonometry function set to the wrong angle mode can therefore produce answers that are far from what you intended.
Angle conversions are especially useful when:
Learning the relationships between the units helps you check whether an answer is reasonable before relying on it.
A convenient way to convert angle units is to use degrees as a reference unit. First convert the input angle into degrees, then convert that degree value into the desired output unit.
The main relationships are:
| Unit | Relationship to degrees |
|---|---|
| Degree | \(1^\circ = 1^\circ\) |
| Radian | \(1\ \text{rad} = \frac{180}{\pi}^\circ\) |
| Gradian or gon | \(1\ \text{gon} = 0.9^\circ\) |
| Turn | \(1\ \text{turn} = 360^\circ\) |
| Arcminute | \(1\ \text{arcmin} = \frac{1}{60}^\circ\) |
| Arcsecond | \(1\ \text{arcsec} = \frac{1}{3600}^\circ\) |
The most common degree-radian formulas are:
For the other supported units, the degree-based conversions are:
After the degree value is known, the reverse formulas give the equivalent values in each unit:
Suppose an angle is \(2\) radians. To convert radians to degrees, multiply by \(\frac{180}{\pi}\):
So \(2\) radians is approximately \(114.591559^\circ\).
This is a useful reminder that radians and degrees use very different scales. A radian is much larger than one degree.
A right angle is \(90^\circ\). Convert it to radians:
So:
This is why many common unit-circle angles are written as fractions of \(\pi\), such as \(\frac{\pi}{6}\), \(\frac{\pi}{4}\), \(\frac{\pi}{3}\), and \(\frac{\pi}{2}\).
A right angle is \(100\) gradians. Since one gradian equals \(0.9^\circ\):
So:
This works because \(400\) gradians make a full \(360^\circ\) rotation.
An angle of \(1.25\) turns means one full rotation plus one quarter rotation:
So:
The angle \(450^\circ\) is coterminal with \(90^\circ\), but it is not the same written measure. Keeping \(450^\circ\) can be important when the number of rotations matters.
One degree contains \(60\) arcminutes and \(3{,}600\) arcseconds. For example, \(30\) arcminutes is:
So:
For arcseconds, \(45\) arcseconds is:
So:
Arcseconds are useful when the angle is too small to describe conveniently in whole degrees.
The primary result is the angle converted to degrees. This gives a familiar reference point, even when the original input is in radians, gradians, turns, arcminutes, or arcseconds.
The other results show the same angle in each supported unit. For example, if the degree result is \(180^\circ\), the equivalent values are \(\pi\) radians, \(200\) gradians, \(0.5\) turn, \(10{,}800\) arcminutes, and \(648{,}000\) arcseconds.
A negative result keeps the sign of the entered angle. In many diagrams, positive angles are measured counterclockwise and negative angles clockwise, but the converter does not decide the physical direction for you. It simply preserves the sign while converting units.
A result greater than one full rotation is not automatically reduced. For example, \(450^\circ\) remains \(450^\circ\), not \(90^\circ\). Those two angles are coterminal, but they can mean different things when the number of full rotations matters.
The radians result may appear as both a decimal value and a simplified multiple of \(\pi\) only when the degree equivalent is an exactly represented safe-integer degree value. For other values, a decimal radian result is usually the most practical format.
Choosing the wrong source unit. The number \(90\) means very different things depending on whether the selected unit is degrees, radians, gradians, turns, arcminutes, or arcseconds. Always match the selected unit to the number you entered.
Typing unit symbols into the value field. Enter only the numeric value. Do not type symbols or words such as \(^\circ\), rad, gon, or turn into the number field.
Entering symbolic pi expressions. Expressions such as \(\pi\), \(\pi/2\), and \(2\pi\) are mathematical notation, not decimal input. To convert \(\frac{\pi}{2}\) radians, enter a decimal approximation such as \(1.5707963268\) and choose radians.
Using commas as thousands separators. A decimal comma may be treated as a decimal point. That means 1,5 is read like \(1.5\), not \(1{,}500\). Avoid thousands separators in the input.
Expecting automatic angle wrapping. An angle such as \(720^\circ\) is not reduced to \(0^\circ\). The result represents the entered rotation amount, including full turns.
Rounding too early. Radian conversions often involve \(\pi\), so decimal values are approximations. Keep enough digits during intermediate work, especially for homework, surveying, engineering, or scientific calculations.
Use angle conversion when you need to move between different ways of describing the same rotation. Common cases include:
For learning math, the most important relationship is usually \(180^\circ = \pi\) radians. For practical measurement, the most useful habit is to write the unit next to every angle until the conversion is complete.
Angle conversion gives equivalent measurements; it does not explain the physical situation behind the angle. A converted value cannot tell you whether an angle was measured clockwise, counterclockwise, from north, from the positive \(x\)-axis, or from another reference direction unless that context is supplied separately.
The supported units are degrees, radians, gradians, turns, arcminutes, and arcseconds. Other formats, such as degrees-minutes-seconds notation, symbolic pi expressions, fractions, and values with unit suffixes, need to be rewritten as decimal numbers before using the calculator.
The converter does not normalize coterminal angles into a fixed interval such as \(0^\circ\) to \(360^\circ\) or \(-180^\circ\) to \(180^\circ\). This is useful when the entered value represents an actual amount of rotation, but it means you may need a separate coterminal-angle calculation if you want a standard-position angle.
Displayed decimals are rounded for readability. Very large or very small finite values may be shown in scientific notation. If converting an otherwise finite input would overflow or underflow any supported result, the calculator reports a range and precision limitation instead of displaying a misleading zero or placeholder. For measurement-critical work, double-check the result, keep appropriate precision, and follow the standards or procedures required for the field.
Decimal values, negative values, signed values, leading decimals such as .5, and scientific notation such as 1e3 can be used. Keep the input numeric: do not include unit symbols, degree signs, pi notation, fractions, or thousands separators.
Multiply the radian value by \(\frac{180}{\pi}\):
For example, \(2\) radians is approximately \(114.591559^\circ\).
Multiply the degree value by \(\frac{\pi}{180}\):
For example, \(90^\circ\) equals \(\frac{\pi}{2}\) radians.
Radians are based on the relationship between a circle's radius and arc length. Because a full circle has circumference \(2\pi r\), one full rotation is \(2\pi\) radians. That is why many common angles can be written neatly as fractions or multiples of \(\pi\).
No. Enter a decimal number instead. For example, use \(1.5707963268\) for \(\frac{\pi}{2}\) radians or \(6.2831853072\) for \(2\pi\) radians.
Yes. A negative value is converted with the negative sign preserved. For example, \(-90^\circ\) converts to \(-\frac{\pi}{2}\) radians, \(-100\) gradians, and \(-0.25\) turn.
They are coterminal because they end in the same direction after full rotations are ignored. They are not the same written measure, because \(450^\circ\) includes one full \(360^\circ\) rotation plus another \(90^\circ\). This calculator keeps the entered rotation amount instead of reducing it automatically.
Arcminutes and arcseconds describe small angles. One arcminute is \(\frac{1}{60}\) of a degree, and one arcsecond is \(\frac{1}{3600}\) of a degree. They are common in fields that need fine angular detail, such as astronomy, mapping, navigation, and surveying.
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