Question

How do astronomers know the distance to moon?

Answer 1

There are many ways to determine the distance, including geometrically, by orbital calculation, and by bouncing a laser beam off it.

Explanation:

Today we know the distance to the moon quite precisely, because astronomers can shoot a laser at the moon, which bounces off reflectors left by Apollo astronauts. by measuring the time between when the beam is sent and the return signal is detected, and knowing the speed of light, the distance can be very precisely determined.
Geometrically the distance can be determined by triangulation - measuring the angle to the moon at the same time from different locations on earth.
The ancient Greeks estimated the distance to the moon by comparing the width of the moon's shadow during a solar eclipse to the Earth's shadow during a lunar eclipse. With enough additional information, such as the diameter of the Earth, the distance can be calculated.
Also, we can determine the orbital period of the moon very precisely, and we can use that to determine the distance using the equations of orbital mechanics.
But since we have lasers, we don't need to do those other things!
Fun facts:
The Moon's orbit is elliptical, getting as close as 357,000 km, and as far away as 407,000 km. The average (semi-major axis) distance is 385,000 km.
The moon is moving away from the Earth at about 4 cm each year!