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What's the name of the measure of apparent (visible) size of a body? e.g. the Sun and the Moon appear to be roughly the same size, despite the Sun being vastly bigger - because the Moon is much closer. What's this apparent size called?

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It's angular diameter (a.k.a angular size or apparent size). This XKCD comic strip shows in an interesting way the angular size of some of the most remarkable objects in the sky:

Comic about apparent size of celestial bodies

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Angular size. That is, the vieving angle that the body spans on sky.

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