Textbooks reluctantly write it as 384,400 km on average. But How on earth did they figure out how far the moon was? Here is an interesting way:
The Lunar Laser Ranging Experiment.
The ongoing Lunar Laser Ranging Experiment measures the distance between the Earth and the Moon using laser ranging.
Lasers on Earth are aimed at retro-reflectors ( device or surface that reflects light back to its source with a minimum of scattering.) planted on the Moon during the Apollo program (11, 14, and 15), and the time for the reflected light to return is determined.
The distance to the Moon is calculated approximately using this equation:
Distance = (Speed of light × Time taken for light to reflect) / 2.
(since distance=speed* time )
The reflected light is too weak to be seen with the human eye: out of 1017 photons aimed at the reflector, only one will be received back on Earth every few seconds, even under good conditions. They can be identified as originating from the laser because the laser is highly monochromatic.
There you have it! The distance between the moon and the earth, and also conclusive proof that astronauts did land on the moon.
Can you test this in your backyard ? Unfortunately no ! You would need highly sensitive detectors and a laser that can shoot 1017 green 532 nm photons per pulse. Not to mention the fancy electronics ! Phew !
Mythbusters try the lunar ranging experiment. ( for real ) )