Degrees to Radians Converter

A degree divides a circle into 360 pieces - a number inherited from Babylonian base-60 counting. A radian takes a different approach: it measures an angle by the arc length it creates on a unit circle. Because the arc of one radian on a radius-1 circle is exactly one unit long, there are 2π radians in a full circumference, which is where the factor 2π ≈ 6.2832 comes from.

Radians matter because calculus only works cleanly in them. The famous limit sin(x)/x → 1 is true only when x is in radians, and the derivatives of sine and cosine lose their awkward scaling factors entirely: in radians, the derivative of sin(x) is exactly cos(x). Switch to degrees and every derivative picks up a factor of π/180.

Most programming languages follow the same convention. JavaScript's Math.sin(), Python's math.sin() and C's sin() all expect radians, so a measured angle of 45° must be converted before it is passed in. That is the single most common source of wrong results in graphics and simulation code.

Conversion in practice: multiply by π/180 to go from degrees to radians, and multiply by 180/π to come back. For quick estimates, one radian is about 57.3°, so 1 rad ≈ 57.3°, 0.1 rad ≈ 5.73°, and a small angle in radians is approximately its sine and tangent - the small-angle approximation used constantly in physics and engineering.

Watch out for two neighbouring units: gradians (also called grads or gon), where a full turn is 400 grads, and turns (revolutions), where a full turn is 1. A value of 100 in a system is 90° if it is gradians and 360° if it is turns, so check which unit your instrument or software is reporting.