This angle converter converts degrees, radians, gradians, turns (revolutions), arcminutes and arcseconds in both directions — for math class, machining, astronomy and every settings menu that demands radians.
Why So Many Ways to Slice a Circle?
Degrees (360 per circle) descend from Babylonian astronomy and divide beautifully (by 2, 3, 4, 5, 6, 8, 9, 10, 12…). Radians (2π per circle) are mathematics’ native unit — the angle whose arc equals the radius — and the only unit in which calculus formulas stay clean. Gradians (400 per circle) were the French Revolution’s decimal attempt, surviving in European surveying. Turns serve rotation machinery, and arcminutes/arcseconds (60 per degree, 3,600 per degree) serve astronomers and navigators. Percent-grade versions of tilt live on our Slope Calculator. Part of our Conversion Calculators collection.
The Benchmarks Worth Knowing Cold
| Degrees | Radians | Fraction of a turn | Name |
|---|---|---|---|
| 30° | π/6 = 0.524 | 1/12 | — |
| 45° | π/4 = 0.785 | 1/8 | — |
| 90° | π/2 = 1.571 | 1/4 | Right angle |
| 180° | π = 3.1416 | 1/2 | Straight angle |
| 270° | 3π/2 = 4.712 | 3/4 | — |
| 360° | 2π = 6.283 | 1 | Full turn |
Worked Examples (Verified)
- 180° = π radians (3.1416) = 200 gradians = half a turn — the unit systems’ Rosetta stone
- 1 radian = 57.30° — bigger than most people guess, which is why a “1 rad” machining spec deserves a second look
- The Moon’s width in the sky ≈ 31 arcminutes = 0.52° — astronomy’s small-angle world runs on the ′ and ″ units in this converter
Related Tools
- Slope Calculator — angles as rise/run and percent
- Roof Pitch Calculator — degrees in the wild
- Conversion Calculators — the full collection
Frequently Asked Questions
How do I convert degrees to radians?
Multiply by π/180: 180° = π rad ≈ 3.1416, 90° = π/2 ≈ 1.5708, 45° = π/4 ≈ 0.7854. Reverse: radians × 180/π (1 rad = 57.30°).
Why do mathematicians prefer radians?
Because the formulas become natural: arc length = rθ, sin(x) differentiates to cos(x), and series expansions work only in radians. Degrees are a human convention; radians are geometry’s own scale.
What is a gradian used for?
Surveying (mainly European) and some artillery systems: 400 gon per circle makes right angles exactly 100 gon, decimal-friendly. Mils (6,400 per circle in NATO use) serve similar military purposes and are included here.
How many arcseconds are in a degree?
3,600 (60 arcminutes × 60 arcseconds) — the Babylonian base-60 inheritance that also structures our clocks. A parsec is defined through one arcsecond of parallax.
What angle is a full turn in radians?
2π ≈ 6.2832 rad. Half a turn (straight angle) is π; a quarter turn (right angle) is π/2.
Is this angle converter free?
Yes — free, instant, no sign-up.