I How Aristarchus found moon's diameter?

You could look him up on the Internet. There are several fine articles there which discuss Aristarchus and his work.

 

Similar triangles?
PF will respond so much better when you appear to have put some effort in, yourself.
What have you actually learned from that Wiki article?
I have spent less than five minutes on the recommended Wiki article and I found:
"The error in this calculation comes primarily from the poor values for x and θ. The poor value for θ is especially surprising, since Archimedes writes that Aristarchus was the first to determine that the Sun and Moon had an apparent diameter of half a degree. This would give a value of θ = 0.25, and a corresponding distance to the moon of 80 Earth radii, a much better estimate. The disagreement of the work with Archimedes seems to be due to its taking an Aristarchos statement that the lunisolar diameter is 1/15 of a "meros" of the zodiac to mean 1/15 of a zodiacal sign (30°), unaware that the Greek word "meros" meant either "portion" or 7°1/2; and 1/15 of the latter amount is 1°/2, in agreement with Archimedes' testimony."

Is this a homework assignment?

 

Ouch. That was hardly an explanation - not one that could satisfy me, in any event.
But you really have to hand it to those ancients who managed to measure some of those astronomical angles and distances a lot better than you or I would do, using tools from our shed and some bits of good quality timber!