384,400km. However, since the Moon’s orbit is not perfectly circular (because orbits like that only exist in theory), this distance is only an average. At its closest point (known as perigee), the Moon is 363,104 km from Earth and at its furthest point (known as apogee)it is 405,696 km away. Incidentally, these distances are from the centre of the Earth to the centre of the Moon. If you want the distance from surface to surface, you’ll have to take into account the Earth’s radius of about 6400km, and the Moon’s radius of 1700km (in other words, subtract 8100km from the figure).
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