The distance to the moon is solidly known. If you have a steady enough aim (Not your hand, some kind of system), a laser, a receiver, and good timing software for it, you can calculate using the speed of light as a known constant and the return time to get the distance. We have done this using the Lunar Rangefinding Receivers, left by the Apollo astronauts.
Amateur radio operators have also calculated the distance to the moon in their "Moon Bounce" experiments, using the speed of radio waves as a constant and the return time to get the distance. They have done this by transmitting radio signals to the moon, "bouncing" them off the moon's surface , and noting when they were received back on the earth.
The results compared with the known distance and it was not 3,000 miles. On one example the distance was calculated to be 238,150 miles.