Angle Converter
Convert between degrees, radians, gradians, turns, arcminutes and arcseconds with visual angle indicator
Visual Angle
All Conversions
degrees
90.0000
radians
1.5708
gradians
100.0000
turns
0.2500
arcmin
5400.00
arcsec
324000.00
Common Angles
Right angle
90deg
1.5708 rad
Straight line
180deg
3.1416 rad
Full circle
360deg
6.2832 rad
Full turn
360deg
6.2832 rad
Half turn
180deg
3.1416 rad
Quarter turn
90deg
1.5708 rad
Angle Units
- Degrees: 360 per full circle (most common)
- Radians: ~6.28 per full circle (mathematics, physics)
- Gradians: 400 per full circle (surveying, some engineering)
- Turns: 1 = full rotation
- Arcmin/Arcsec: for precise angles (astronomy, surveying)
Angle Measurement Units
Three main units: degrees (Β°, 360 in a circle), radians (rad, 2Ο in a circle), gradians (gon, 400 in a circle). Conversions: degrees Γ Ο/180 = radians. Degrees Γ 10/9 = gradians. So 90Β° = Ο/2 rad = 100 gon. Used in: geometry, trigonometry, surveying, navigation, astronomy.
Most everyday work uses degrees - intuitive and what calculators default to. Mathematics (especially calculus) uses radians for cleaner formulas. Engineering surveying sometimes uses gradians (especially in France/Germany). Astronomy uses degrees, arc minutes (1Β° = 60'), arc seconds (1' = 60").
Common Angle Conversions
| Degrees | Radians | Gradians |
|---|---|---|
| 0Β° | 0 | 0 gon |
| 30Β° | Ο/6 β 0.524 | 33.33 gon |
| 45Β° | Ο/4 β 0.785 | 50 gon |
| 60Β° | Ο/3 β 1.047 | 66.67 gon |
| 90Β° | Ο/2 β 1.571 | 100 gon |
| 180Β° | Ο β 3.142 | 200 gon |
| 270Β° | 3Ο/2 β 4.712 | 300 gon |
| 360Β° | 2Ο β 6.283 | 400 gon |
Frequently Asked Questions
Why are circles 360Β°?
Inherited from Babylonian mathematics (~3000 BC). 360 has many divisors (1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360), making it convenient for fraction-based geometry. Pre-decimal era preferred convenient fractions.
Should I learn radians?
If you do calculus, physics, engineering, or programming with trig functions: yes, radians become natural. If you only do everyday geometry: degrees are fine. Both are correct; mathematical work prefers radians for cleaner derivatives and integrals.