The Angle Converter converts plane angles among degrees, radians, gradians (gon), and turns. Enter a value in any supported unit and read the equivalents with the factor line and substituted arithmetic. A full turn is 360 degrees, 2π radians, 400 gradians, or 1 turn, so every conversion on the page is a scaled fraction of that circle.
Arcminutes and arcseconds appear when fine angular measure is needed. For latitude and longitude written as degrees, minutes, and seconds on a map, use the Decimal Degrees to DMS Calculator instead of treating DMS as a separate angle unit family here.
Convert between units of angle
The Angle Converter accepts a numeric angle and maps it through a shared turn-based (or radian-based) internal value, then formats every selected target unit. Degrees remain the everyday unit for navigation and school trigonometry. Radians are the SI unit for mathematics and calculus.
Gradians divide a right angle into 100 parts. Turns measure full rotations as pure fractions of a circle.
Pick source and target units, type the value, and the Angle Converter recalculates live. Cascade mode can list degree, radian, gradian, and turn columns from one entry so 90 degrees appears as π/2 radians, 100 gradians, and 0.25 turns in the same panel.
Because the relationships are exact ratios of a turn, converting 180 degrees to radians uses π, not a rounded stand-in, until display precision truncates the decimal form. Exact π expressions stay available when the Angle Converter reports a clean multiple of π.
Convert degrees to radians
Degrees convert to radians by multiplying by π/180. One full circle is 360 degrees and 2π radians, so each degree is π/180 radians. Ninety degrees become π/2 radians, and 180 degrees become π radians.
radians = degrees × (π / 180)Worked case: convert 60 degrees to radians.
1. Multiply by π/180. 60 × π/180 = π/3 radians.
2. Decimal check. π/3 ≈ 1.04719755 radians when π ≈ 3.14159265.
The Angle Converter shows both the exact π form when the result is a simple fraction of π, and the decimal expansion for calculator or code use. Survey and CAD workflows often keep degrees; programming libraries often expect radians for sine and cosine inputs.
Convert radians to degrees
Radians convert to degrees by multiplying by 180/π. That is the inverse of the degree-to-radian factor. One radian equals about 57.29577951 degrees, because 180/π ≈ 57.29577951.
degrees = radians × (180 / π)Worked case: convert π/4 radians to degrees.
1. Multiply by 180/π. (π/4) × (180/π) = 45 degrees exactly.
2. Decimal input. 1 radian × (180/π) ≈ 57.2958 degrees.
The Angle Converter cancels π symbolically when the input is an exact multiple of π, so π/6 becomes 30 degrees without floating-point drift in the reported exact form. Decimal radian inputs still use full π precision internally before display rounding.
Express radians in terms of π
Many textbook answers prefer radians written as a multiple of π rather than a long decimal. Half a turn is π radians, a quarter turn is π/2, a sixth of a turn is π/6, and a full turn is 2π.
The Angle Converter can present those exact forms when the degree or turn input lands on a clean fraction of the circle.
| Degrees | Exact radians | Decimal (approx.) |
|---|---|---|
| 30 | π/6 | 0.5236 |
| 45 | π/4 | 0.7854 |
| 60 | π/3 | 1.0472 |
| 90 | π/2 | 1.5708 |
| 180 | π | 3.1416 |
| 360 | 2π | 6.2832 |
Gradians fit the same circle: 100 gon = 90 degrees = π/2 radians. One turn equals 400 gradians. Mixing gon with degrees without converting first is a common surveying error; the Angle Converter keeps the 400-gon full circle explicit beside the 360-degree circle.
Convert gradians and turns
Gradians convert to degrees by multiplying by 0.9, because 360/400 = 0.9. Degrees convert to gradians by multiplying by 400/360 = 10/9. A right angle is 100 gradians; a straight angle is 200 gradians. The Angle Converter shows those equivalents beside degree and radian columns in cascade mode.
Turns are fractions of a full rotation. One quarter turn is 0.25 turns = 90 degrees = π/2 radians = 100 gradians. Machine and motor contexts often quote revolutions as turns; school trigonometry usually stays in degrees or radians. Entering 0.5 turns should read 180 degrees and π radians on the result panel.
Arcminutes and arcseconds refine a degree: 1 degree = 60 arcminutes = 3600 arcseconds. When a map angle is already written as degrees-minutes-seconds for a coordinate, prefer the Decimal Degrees to DMS Calculator. On this page, fine units still sit inside the plane-angle family when selected in the unit list.
FAQ
How do I convert degrees to radians?
To convert degrees to radians, multiply degrees by π/180. For example, 180 × π/180 = π radians. The Angle Converter multiplies by that factor and can show the exact π form when the angle is a clean fraction of a turn.
How do I convert radians to degrees?
To convert radians to degrees, multiply radians by 180/π. For example, π/2 × 180/π = 90 degrees. The Angle Converter uses the same inverse factor for every radian input.
What is 180 degrees in radians?
One hundred eighty degrees equal π radians, about 3.1416 in decimal form. The Angle Converter reports both the exact π value and the rounded decimal for that half-turn angle.
What is a gradian?
A gradian (also called a gon) is an angle unit with 400 gradians in a full turn, so a right angle is 100 gradians. The Angle Converter converts gradians to degrees by multiplying by 0.9, because 360/400 = 0.9.
How many degrees are in one radian?
One radian equals 180/π degrees, about 57.2958 degrees. The Angle Converter applies that factor when the source unit is radians and the target is degrees.
What is a turn in degrees?
One turn equals 360 degrees, 2π radians, or 400 gradians. The Angle Converter treats a turn as a full rotation so fractional turns such as 0.25 map to 90 degrees.
Is this the same as the DMS calculator?
No. The Angle Converter changes plane-angle units such as degrees and radians. The Decimal Degrees to DMS Calculator reformats a single degree measure into degrees, minutes, and seconds for coordinates. Use DMS formatting when the need is map-style sexagesimal display.
Why do programming languages use radians?
Trigonometric functions in most math libraries take radians because calculus and the SI unit system define angular measure that way. Degrees still appear in user interfaces, so the Angle Converter bridges UI degree inputs and radian APIs with the π/180 factor.
Summary
The Angle Converter converts degrees, radians, gradians, and turns using exact circle ratios: 360 degrees, 2π radians, 400 gradians, and 1 turn. Degrees become radians by multiplying by π/180; radians become degrees by multiplying by 180/π. Exact π forms such as π/2 for 90 degrees sit beside decimal results, and cascade views show every unit from one entry.
Coordinate DMS formatting belongs on the Decimal Degrees to DMS Calculator, while this page owns unit conversion for plane angles.