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Before we knew the distance to the moon, how can you get the exact angular degree of it without knowing the distance to it? Obviously you can just guess or assume but how was it first calculated?

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You measure the angle between a line pointed at one limb of the moon and a line pointed at the opposite limb.


Note that "angular size" is an angle. You measure it like any other angle.

Of course, there is a relationship between linear size, angular size, and distance $$ \sin \theta = \frac{l}{d} \;,$$ for $\theta$ the angular size, $l$ the linear size transverse to the line of sight and $d$ the distance. So that knowing any two you can find the third. But in this case angular size is the easiest one to measure.

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