Degrees to Radians

To go from degrees to radians, multiply by π / 180: 180 × π / 180 = π (about 3.1416 rad). For the reverse, multiply by 180 / π. The key reference point: 180 degrees = π radians, a half turn — everything else scales proportionally from there.

A radian is defined geometrically, not arbitrarily: it is the angle at the center of a circle where the arc length equals the radius. Walk a distance of exactly one radius along the circle's edge, and the angle you have swept out, measured from the center, is one radian — about 57.3 degrees. Because a full circle's circumference is 2πr, exactly 2π of those radius-lengths fit around it, which is precisely why a full turn is 2π radians and a half turn is π. The unit is not a human convention layered on top of geometry; it falls directly out of the geometry itself.

Degrees, by contrast, are a historical convention — 360 divisions of a circle, inherited from Babylonian base-60 astronomy, which tracked roughly 360 days in its approximation of a year and split the circle to match. The number stuck for a practical reason that still applies today: 360 has 24 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), letting a circle be split evenly into halves, thirds, quarters, fifths, sixths, eighths, tenths, twelfths and more, all landing on whole numbers. A round number like 100 or 1,000 divides far less cleanly, which is part of why degrees outlasted more "metric" attempts to replace them (France's short-lived 400-gradian circle among them).

Nearly every programming language's built-in trigonometric functions — JavaScript's Math.sin(), Python's math.sin(), and the equivalents in most others — expect the input in radians, not degrees, because radians are what the underlying calculus is built on. Calling Math.sin(90) does not return 1; it returns roughly 0.894, the sine of 90 *radians* (which is over 14 full turns around the circle), not 90 degrees. The fix is always the same: convert first — Math.sin(90 * Math.PI / 180) correctly returns 1. This single mismatch is one of the most common silent bugs in code that touches angles, because it produces a plausible-looking number instead of an obvious error.

Beyond the geometric definition, radians earn their place because calculus works out cleanly only in that unit: the derivative of sin(x) is exactly cos(x) when x is in radians — in degrees, an extra conversion constant (π/180) creeps into every derivative, which is exactly why physics, engineering and computing standardized on radians rather than degrees for anything beyond basic geometry. That standardization is also why this is a genuinely one-way piece of knowledge to carry: degrees remain useful for communicating an angle to a person — "turn the wrench 90 degrees" is intuitive in a way "turn it π/2 radians" is not — while radians take over the moment the angle enters a formula, a derivative, or a line of code. Knowing which context you're in, human-facing or computational, decides which unit belongs on the page.