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Angle Converter
Convert between degrees, radians, gradians and turns.
Babylonian astronomers divided the circle into 360 parts around 4,000 years ago, probably because 360 is close to the number of days in a year and divides evenly by almost everything. That single choice is why your protractor, your compass and your phone’s screen-rotation setting all still speak in degrees.
The four ways to measure a turn
- Degree — one 360th of a full circle. Convenient because 360 divides by 2, 3, 4, 5, 6, 8, 9, 10, 12 and more with no remainder.
- Radian — the angle where the arc length equals the radius. A full circle is 2π radians — about 6.2832 — which is why radians look ugly on paper but make calculus formulas clean.
- Gradian — one 400th of a circle, so a right angle is a tidy 100 gradians. Built for decimal-friendly surveying and mostly forgotten outside that field.
- Turn — one full revolution, equal to 360°, 2π radians, or 400 gradians. Used in engineering when something rotates a known number of times.
Why radians are not optional in mathematics
Degrees are arbitrary — there is nothing mathematically special about 360. Radians are not; they fall directly out of the geometry of a circle, which is why every calculus and physics formula involving sine, cosine or angular velocity is written in radians rather than degrees.
This is also why a calculator set to the wrong angle mode gives wildly wrong answers. Ask for sin(90) in degree mode and you get 1. Ask in radian mode and you get about 0.894, because 90 radians is more than fourteen full turns around the circle.
Angles worth knowing by heart
| Degrees | Radians | Gradians | What it is |
|---|---|---|---|
| 0° | 0 | 0 | No rotation |
| 30° | π/6 | 33.3 | Half of 60° |
| 45° | π/4 | 50 | Half a right angle |
| 90° | π/2 | 100 | A right angle |
| 180° | π | 200 | A straight line |
| 360° | 2π | 400 | A full circle |
Where each unit actually shows up
- Degrees: navigation bearings, protractors, CSS transforms, and every compass you will ever hold.
- Radians: physics, engineering and every programming language’s built-in trig functions, which expect radians by default.
- Gradians: land surveying in parts of Europe, and some scientific calculators still carry a GRAD mode as a leftover.
- Turns: motor and gearbox specifications, where an engineer cares how many complete rotations a shaft makes.
Angle conversion questions
How do I convert degrees to radians by hand?
Multiply the number of degrees by π and divide by 180. So 90° becomes 90 × π ÷ 180, which simplifies to π/2, roughly 1.5708 radians.
Why does my calculator give the wrong answer for sine and cosine?
It is almost always the angle mode. Scientific calculators can be set to degrees, radians or gradians, and mixing the wrong mode with your input produces a number that looks plausible but is completely off. Check the mode indicator first whenever a trig result looks strange.
What is a gradian actually used for today?
Mostly historical French surveying instruments and a handful of European land-registry systems that adopted the decimal circle in the metric reforms of the 1790s. Outside those niches it rarely appears.
Is 2π radians exactly 360 degrees?
Yes, exactly, by definition — both describe one full rotation. Because π is irrational, the radian value never terminates in decimal form, which is the main reason degrees remain more convenient for everyday, non-mathematical use.
Why do compass bearings run 0 to 360 instead of using negative numbers?
Because a bearing describes a direction, not a signed rotation, and running clockwise from north with no negative values avoids ambiguity at sea or in the air. A bearing of 270° is unambiguous; −90° would require everyone to agree on a rotation direction first.
