What, if anything, is a square radian?

"rad(ians)" is just another word for the number $1$, a unit-less unit. An angle is some part of a full rotation at $2\pi$, a unitless number. However, to tell that any number stands for an angle and not some amount of apples or cake, there is the habit to add "radians" to it.

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Geometry

Classical Mechanics

Circles

Dimensional Analysis

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