When the moon is half full, the sun is just under 90 degrees away in the sky. This is how we know the moon is close by. You can use the law of sines to find out

When the moon is half full, the sun is just under 90 degrees away in the sky. This is how we know the moon is close by. You can use the law of sines to find out how many moon-earth distances are between Earth and then sun. Then, if you have a few friends who know what day it is in different countries, you can find out how large the moon's shadow is on the Earth. The rest is just math.