Convert angles between degrees, radians, gradians, turns, arcminutes and arcseconds — with exact factors.
What Is an Angle Converter?
An angle converter moves between angle units: 180° = π radians = 3.1416 = 200 gradians = 0.5 turns. Degrees are the everyday unit (360 to a circle, from Babylonian astronomy); radians are the mathematician’s unit (the angle whose arc equals the radius — calculus only works in radians); gradians split the right angle into 100 (a French Revolutionary idea, still used in some surveying); turns count full revolutions.
How to Use
- Enter the value.
- Pick the “from” unit.
- Pick the “to” unit.
- Click “Convert Angle”.
Formula
Convert to degrees first (value × from-factor), then divide by the to-factor.
Worked Example
180° → radians: 180 ÷ 57.29577951308232 = 3.1416 rad (= π). And 90° → gradians: 90 ÷ 0.9 = 100 grad.
How It Works, In Plain Words
Every unit in the two dropdowns is stored as a single number: how many degrees one unit of it contains. A degree is 1, a radian is 57.29577951308232, a gradian is 0.9, a turn is 360, an arcminute is 1/60 of a degree and an arcsecond is 1/3600. Converting in two hops keeps every pair consistent: first multiply your value by its from-factor to land on plain degrees, then divide by the to-factor to step into the unit you want. The worked example shows the long way round on purpose: 180° becomes radians by dividing 180 by 57.29577951308232 to get 3.1416.
The same two hops explain the page’s other check figure: 90° in gradians is 90 divided by the gradian factor 0.9, giving exactly 100 grad — a quarter turn, since a full turn is 360° and a right angle is a quarter of one. Radians skip the decimal clutter for the famous angles because they are defined by the circle itself: the angle whose arc equals the radius, so 180° lands exactly on π. The result is shown to four decimal places, which is why 3.1416 is a rounded display of π rather than the whole value.
Real-World Uses
A maths student hits the page’s own trap from the mistakes list: the homework is in degrees but the calculus needs radians, so 180° becomes 3.1416 rad before any differentiating starts. A surveyor working from older European plans reads angles in gradians and converts 100 grad to 90° to set out a right angle that matches the drawings. A navigator or astronomer deals in the tiny end of the scale, where positions come in arcminutes and arcseconds — 1/60 and 1/3600 of a degree — and converts them into decimal degrees a chart or spreadsheet can use. A mechanic or cyclist counting revolutions uses turns instead: half a turn is 0.5 turns, the same 180° the converter opens with. In each case the value never changes — only the language it is expressed in, which is the whole point of converting rather than recalculating.
Benefits
- Six units including gradians and turns
- Radians exact via π
- Both directions in one tool
- Maths, surveying and navigation friendly
Features of Our Tool
- Instant results in your browser
- No sign-up, no download, 100% free
- Works on mobile, tablet and desktop
- Clear step-by-step explanation below the tool
- Private — your entries never leave your device
Common Mistakes
- Doing calculus in degree mode — trig derivatives assume radians
- Confusing gradians with degrees (100 grad = 90°)
- Forgetting arcminutes/arcseconds are 1/60 and 1/3600 of a degree
Common Mistakes to Avoid
Mode confusion starts the list on this page: trig derivatives assume radians, so differentiating in degree mode gives answers that are quietly wrong, not obviously wrong. Gradians cause the mirror-image error — 100 grad looks like a degree-scale number but is 90°, so a layout set out from it leans. The small units trip people differently: arcminutes and arcseconds are sixtieths, not hundredths — 1/60 and 1/3600 of a degree — so reading them as decimals skews a position. Finally, rounding: the tool displays four decimal places, so π appears as 3.1416; carrying that rounded figure into a long chain of further calculations lets small errors accumulate, so keep the full result where precision matters.
FAQ
How many radians is 180 degrees?
3.1416 rad (= π).
What is a gradian?
1/100 of a right angle: 100 grad = 90°.
How many degrees in a turn?
360°.
Why do mathematicians use radians?
Because calculus formulas for trig functions are simplest in radians — the derivative of sin(x) is cos(x) only in radians.
What are arcminutes and arcseconds?
1/60 and 1/3600 of a degree — used in astronomy and navigation for tiny angles.
How many arcseconds are in one degree?
3,600. An arcsecond is 1/3600 of a degree, and an arcminute is 1/60 — the two small units this converter includes for astronomy and navigation work.
Can I convert radians back to degrees?
Yes — pick Radian as the from unit and Degree as the to unit. The two-hop method runs in both directions, which is why 3.1416 rad returns 180°.