The distance between the earth and the moon was first estimated by the Greek astronomer, Aristarchus, around the third century B.C. by observing a lunar eclipse.
By "shooting" a laser at a reflector placed on the moon and measuring the time it takes the light to return back to Earth. By having two people measure where the moon is in the sky at the same time,...
The moons distance from Earth varies during its orbit, from around 405,696 km or 252,088 miles at its furthest (Apogee) to 363,104 km or 225,622 miles at its closest (Perigee).