One thing I've been curious about while learning Chinese is the use of 月 in many 汉字 representing anatomical features (for example, 脸， 腰， or 腿).
What is the etymology of the use of 月 as a radical?
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@Stan very right! The distinction sometimes made between the different origins of what today (and has been so for thousands of years) is indistinguishable from 月 is entirely misguided, artificial, and continues to fail. Blatantly, the Ministry of Education tries to prescribe a somewhat boiled-down version of the 'Original Sin' (that was started during the Tang dynasty and taken up again by the editors of the Kangxi dictionary) and they fail. Read more about this in my lengthy answer here: https://chinese.stackexchange.com/a/12455/3674.
To answer the OP more constructively in broad terms: When Clerical script developed from Seal script sometime before the First Emperor of Qin, the goal was to find a way of writing that was easier and faster. Consequently, many rounded strokes were simplified to straighter lines, many components were written with a simplified outline, and some components were given an abbreviated form when appearing as a 'side-element' (偏旁); this is why we today have such forms as 氵 for 水, 亻 for 人, 灬 for 火 and so on.
In the latter case, some components that differed in Seal script were written with the same abbreviated form as a side-element, and this happened to 月, 肉, 舟, and 丹 (and some others, it would seem) that ended up as 月. Likewise, 舌 and two other components were confounded (or unified, depending on your point of view), which is why the readings of 活, 括 etc are not from 舌 (活 is explained as '从水,昏声'—no mention of 舌); 入 and 人 were not distinguished; 灬 sometimes goes back to 火 and sometimes to something entirely different (as in 馬 and 熊), and so on.