Angle Converter
Convert between degrees, radians, and gradians.
Unit
Angle Converter
Generated on April 24, 2026
Full turns: 0.25
The shaded sector shows the angle measured counter-clockwise from the dashed 0° baseline.
?What is the Angle Converter?
An angle converter moves between degrees (familiar everyday unit), radians (the math and physics native unit), gradians (surveying and some European engineering), turns (clean for rotation problems), and minutes/seconds of arc (astronomy and precision navigation). While a full circle is 360 degrees, it is also 2π radians, 400 gradians, and exactly 1 turn. Mixing these up causes silent bugs in code — for example, most programming language trigonometric functions (sin, cos, tan) expect radians, not degrees.
The Formula
A radian is defined as the angle at which the arc length equals the circle's radius. That is why a full circle is exactly 2π radians — its circumference (2πr) divided by r. Radians are the natural unit in calculus: d/dx(sin x) = cos x only when x is in radians. Degrees (360 per circle) originate in Babylonian astronomy, which used base-60 math and mapped nicely to the 360-day year approximation. Gradians (400 per circle) were a French Revolutionary metric reform that never fully caught on except in surveying and some CAD software.
Practical Examples
90° equals π/2 radians, or 100 gradians — the right angle used everywhere from construction to robotics.
180° equals π radians (≈ 3.14159) — a straight line and the famous constant in one.
One full turn is 360°, or 2π radians, or 400 gradians — any of which can describe a single rotation.
1 degree equals 60 arc-minutes (60′), and 1 arc-minute equals 60 arc-seconds (60″) — used in astronomy for fine angular measurement.
Earth's rotation is 15° per hour (360° ÷ 24), which equals 0.25° per minute — the basis of timezone width.
A GPS position accurate to 1 arc-second corresponds to about 30 meters on the ground at the equator.