The Moon

Using trig to measure the distance, was done by Hipparchus in 129 BC, www.phy6.org/stargaze/Shipparc.htm
He used a location where there was a total eclipse of the sun and then the distance from that location that the eclipse was 4/5 full.



He concluded the distance to the moon was between 62 and 73 earth radii, not too far from current measurements of 60 earth radii.


All this stuff was solved by Newton around 1687, and described earlier by Kepler, the relationship between a planet's orbital period and distance from the sun is well known and understood, see Kepler's third law.
Orbits under an inverse square law force field ( gravity ) are always conic sections, thus some orbits are highly elliptical ( comets ) and some nearly circular ( planets ). These calculations work, and enable precise predictions. At the present time man made spacecraft have visited all the planets, except pluto.

Back to the moon, I would argue the distance is known with mm precision because of the laser ranging work that's been done, bouncing signals off the retro-reflector left by Apollo has enable EME distances to be established that the moon is in fact drifting away at 3.8 cm per year.