Angle Converter (degrees, radians, gradians, turns)
Type an angle value, pick the source unit, and instantly see all 6 equivalent angle representations. Useful for math, physics, navigation, surveying, and CSS.
- Degrees (°)90
- Radians (π)1.570796
- Gradians (gon)100
- Turns (revolutions)0.25
- Arcminutes (')5,400
- Arcseconds (")324,000
How it works
Six angle units explained
Degrees (°): full circle = 360°. Most common everyday unit. Right angle = 90°, half-turn = 180°. Used in geometry, navigation, physics introductions, daily speech.
Radians (rad): full circle = 2π ≈ 6.283 rad. SI/scientific unit; defined as the arc length divided by radius for a unit circle. All math and physics formulas use radians (sin, cos, tan natively expect radians in most programming languages).
Gradians (gon, grad): full circle = 400 grad. Right angle = 100 grad — the appeal: decimal-friendly, used in surveying and some European engineering contexts.
Turns: full circle = 1 turn. Half-turn = 0.5. Used in physics for cyclic motion and in CSS (transform: rotate(0.25turn) = 90°).
Arcminutes ('): 1° = 60 arcmin. Used in navigation (1 nautical mile ≈ 1 arcmin of latitude) and astronomy.
Arcseconds (''): 1° = 3600 arcsec, 1 arcmin = 60 arcsec. Astronomy uses arcseconds extensively (parallax, telescope resolution).
Common conversions to memorize
180° = π rad ≈ 3.14159 rad. The single most useful conversion in math and physics. Most other conversions follow from this.
90° = π/2 rad ≈ 1.5708 rad (right angle).
60° = π/3 ≈ 1.047 rad. 45° = π/4 ≈ 0.785 rad. 30° = π/6 ≈ 0.524 rad. These three are the 'special angles' from trigonometry.
1° = 60 arcmin = 3600 arcsec. 1 arcmin = 60 arcsec.
1 turn = 360° = 2π rad = 400 grad.
Where each unit is preferred
Programming and physics: radians. JavaScript's Math.sin, Math.cos, Math.tan all expect radians. Convert from degrees via × Math.PI / 180.
Education and everyday: degrees. Most students learn trigonometry in degrees. Construction, navigation, and casual measurement all use degrees.
Engineering surveys (Europe): gradians. A 100-gradian right angle simplifies decimal arithmetic in field calculations.
Astronomy: arcminutes and arcseconds for sky positions. Parallax distances are quoted in arcseconds.
CSS animations: degrees, radians, or turns. transform: rotate(45deg) and rotate(0.125turn) are equivalent — pick whichever reads more naturally.
Frequently asked questions
›How do I convert degrees to radians manually?
Multiply by π/180. So 180° × π/180 = π rad. Or remember: 180° = π, then scale: 90° = π/2, 45° = π/4, 60° = π/3.
›Why are radians the SI unit?
Because they're the natural choice for circular motion math: arc length = radius × angle (in radians), but only if angle is in radians. All physics formulas are simpler in radians.
›What's a gradian (or gon) used for?
Surveying and some European engineering. The decimal-friendly 100-gradian right angle simplifies field arithmetic. Mostly a niche unit today.
›Why 360 degrees in a circle?
Historical, from Babylonian base-60 mathematics around 2000 BCE. 360 has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20, etc.) which makes it convenient for fraction-based arithmetic without decimals.
›How accurate is the calculator?
Floating-point precision (about 15-17 significant digits). Round-trip conversions (degrees → radians → degrees) preserve the input to that precision.
›Can I convert negative angles?
Yes. -90° = -π/2 rad = -100 grad = -0.25 turn. The math is fully signed.
›What about angles greater than a full turn?
Supported. 720° = 2 turns = 4π rad. The calculator doesn't normalize to [0, 360) — to do that, use modulo (e.g., x mod 360 for degrees).
›Does the data leave my browser?
No. Conversion runs locally; nothing is sent to a server.
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