Why Radians Are the "Natural" Angle Unit
Degrees divide a full circle into 360 arbitrary parts (a convention traced back to ancient Babylonian base-60 mathematics), while radians define an angle based on arc length relative to radius — a full circle is exactly 2*pi radians, a value that falls directly out of the circle's own geometry rather than an arbitrary human choice. This is exactly why radians, not degrees, are the standard unit in calculus and physics: derivatives of trigonometric functions only take their simple, clean form (like the derivative of sine being cosine) when the angle is measured in radians — using degrees introduces an extra, awkward scaling constant into those same formulas.
Gradians (also called gons) divide a full circle into 400 parts instead of 360 — a metric-inspired alternative designed so a right angle is a clean 100 gradians, though it never achieved wide adoption outside specific surveying and some European engineering contexts.
Arcminutes and arcseconds subdivide a degree the same way minutes and seconds subdivide an hour (60 arcminutes per degree, 60 arcseconds per arcminute) — a precision-oriented unit still used in astronomy and navigation, where fractional degrees need finer resolution than decimal degrees conveniently provide.