'Degree' is Latin for 'a step' — temperature, angles, and diplomas all measured one step at a time.
A unit of measurement for angles or temperature; an academic rank conferred by a university; the extent, measure, or scope of an action, condition, or relation; a step in a scale or series.
From Old French degré (a step, stair, degree), from Vulgar Latin *dēgradus, a compound of Latin dē- (down) and gradus (step, pace, rank). Gradus derives from the PIE root *gʰredʰ- (to walk, go, step), which produced Latin gradī (to step, walk — hence English grade, gradient, gradual, graduate, aggression, congress, digress, egress, ingredient, progress, regress, transgress), and Lithuanian gridýti (to go, wade). The prefix dē- comes from PIE
There are 360 degrees in a circle — a number chosen by the ancient Babylonians, who used a base-60 (sexagesimal) number system. 360 is approximately the number of days in a year and is evenly divisible by 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, and 180 — making it extraordinarily convenient for subdivision. Each degree is one 'step' in the circle, and the choice of 360 steps