Use the free Angle Converter below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
Three Ways to Measure a Turn
An angle can be measured in degrees, radians or gradians, and the three units describe the same turn in different numbers. A full turn is 360 degrees, 2π radians, or 400 gradians. A half turn is 180 degrees, π radians, or 200 gradians. A right angle is 90 degrees, π/2 radians, or 100 gradians. The units differ only in what they count as “one”: degrees split a turn into 360 parts, gradians into 400, and radians measure the turn by the arc length on a unit circle.
Degrees are the everyday unit: maps, protractors, construction and navigation all use them. Radians are the mathematician’s unit, because calculus formulas for sine and cosine only work cleanly in radians. Gradians, also called gons, are the surveyor’s unit in some countries, chosen because 400 parts make right angles into round hundreds. Knowing all three lets you read any source without converting by hand.
The Conversion Rules
The conversions are simple multiplications. Degrees to radians: multiply by π/180. Radians to degrees: multiply by 180/π. Degrees to gradians: multiply by 10/9. Gradians to degrees: multiply by 0.9. Radians to gradians: multiply by 200/π. Every conversion passes through the full-turn identity 360° = 2π rad = 400 gon, so if you remember that one line, you can rebuild any conversion factor from it.
Why does π appear in angle conversion at all? Because a radian is defined by arc length: one radian is the angle that cuts an arc exactly one radius long on a circle. A full circle has circumference 2πr, so a full turn is 2π radians by definition. The π is not an arbitrary constant; it is the geometry of circles showing through the units.
How to Use This Converter
Enter an angle, pick its unit, and press Convert angle. The tool shows the same angle in all three units, plus the key facts: 180° = π rad = 200 gon and 360° = 2π rad = 400 gon. Enter 180 degrees and you get about 3.1416 radians and 200 gradians. Enter π radians, about 3.1416, and you get back 180 degrees.
Negative angles and angles above a full turn are handled without complaint. An angle of 450 degrees converts to 5π/2 radians, about 7.8540, and 500 gradians. The tool does not reduce angles to a 0–360 range, because in many problems the number of turns matters: a wheel that has spun 720 degrees has done two full turns, and reducing it to 0 would lose that information.
A Worked Example: Reading a Textbook
A physics textbook says a pendulum swings through π/3 radians. How many degrees is that? Multiply by 180/π: (π/3) × (180/π) = 60 degrees. Enter 1.0472 radians in the tool, the decimal for π/3, and the degrees read about 60. The π cancels in the hand calculation, which is why textbook angles are often written as fractions of π: the conversion back to degrees is then a single division.
Another common case: a surveying document gives 125 gradians. Multiply by 0.9 to get 112.5 degrees. Enter 125 gradians in the tool and all three forms appear, including about 1.9635 radians. Gradians feel unfamiliar at first, but the 0.9 factor makes them quick to convert once you know it.
Common Mistakes
The most common mistake is feeding radians into a calculator set to degrees, or the reverse. Sine of 90 radians is not 1; sine of 90 degrees is. If a trigonometry answer looks wildly wrong, check the mode before anything else. The second mistake is confusing gradians with degrees: 100 gradians is a right angle, not a nearly-straight one.
A third mistake is rounding π too early. Using 3.14 instead of the full value introduces a small error that grows when the angle is multiplied further. The tool keeps full precision internally and only rounds the display, so its answers are as accurate as the input allows.
Try It Yourself
Start with 180 degrees and confirm π radians and 200 gradians. Then 90 degrees for π/2 and 100 gradians. Then 100 gradians back to 90 degrees. Then 360 degrees for 2π and 400 gradians. Those four checks tie all three units together. Use the tool as often as you like; every calculation runs in your browser, and nothing you type is sent anywhere.