Degrees per Radian

One radian is the angle that subtends an arc equal to the radius, so a full circle contains 2π of them, and each is 180/π = 57.29578 degrees. The conversion is exact by definition — the degree is defined as exactly π/180 rad — so this constant carries no uncertainty despite its endless decimal expansion. Surveyors and navigators know it as 57.3, and the small-angle rule of thumb that one milliradian subtends one metre at a kilometre follows straight from it.

The radian is the natural unit for calculus: d(sin x)/dx = cos x only when x is in radians, and every series expansion in trigonometry assumes it. The degree is the older convention, inherited from Babylonian base-60 arithmetic and a 360-day approximation to the year, kept because 360 divides evenly by 24 different integers where 2π divides by nothing. Almost every trigonometric bug in production code is a degree fed into a function that wanted radians.