Rotation
Convert angles between degrees, radians, gradians, and more.
Result
0.017453Radian
Base unit: degree. All conversions calculated from this common reference.
An angle converter is a tool that translates angular measurements from one unit to another. The most common conversions are between degrees (360 per full rotation), radians (2π per full rotation, used in calculus and physics), gradians (400 per full rotation, used in surveying), and turns (1 = one complete rotation).
The converter uses fixed mathematical relationships between angle units. Since all units measure the same rotation, the tool multiplies your input by the appropriate conversion factor. For example, degrees to radians multiply by π/180; radians to degrees multiply by 180/π. Enter any angle in any unit and the tool instantly calculates all equivalent values.
Degrees to Radians: radians = degrees × (π/180); Radians to Degrees: degrees = radians × (180/π); Degrees to Gradians: gradians = degrees × (400/360); Gradians to Degrees: degrees = gradians × (360/400); Degrees to Turns: turns = degrees / 360π (pi) ≈ 3.14159 is the mathematical constant relating circles to their diameter. The conversion factors come from the fact that 360° = 2π radians = 400 gradians = 1 full turn.
| Input | Result | Notes |
|---|---|---|
| 45 degrees | 0.785 radians, 50 gradians, 0.125 turns | 45° is 1/8 of a full rotation, a common angle in geometry and carpentry |
| π radians (3.14159) | 180 degrees, 200 gradians, 0.5 turns | π radians equals a straight angle, the most important angle in trigonometry |
| 100 gradians | 90 degrees, 1.571 radians, 0.25 turns | Gradians are used in surveying; 100 gradians = a right angle |
Radians are the natural unit for calculus and trigonometry because they relate arc length directly to the radius: arc length = radius × angle in radians. This makes derivatives and integrals much cleaner.
Gradians (also called gons) divide a circle into 400 equal parts instead of 360. Surveyors and some European engineers use gradians because 100 gradians = a right angle, making decimal calculations cleaner.
Yes, negative angles represent rotation in the opposite direction (clockwise instead of counterclockwise). The conversion formulas work identically for negative values.
The tool uses full floating-point precision (typically 15-17 significant digits), well beyond the precision needed for any real-world application.
An angle measures the separation between two lines or directions (0–360°). A rotation is motion through that angle. The units are identical; the tool converts both.
Ancient Babylonians chose 360 because it divides evenly by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, and 18, making fractions of a circle easy to calculate by hand.