Aristarchus calculation of the "size" of the sun

The full video may explain this but if someone doesn't want to watch that Wikipedia says that Aristarchus' calculations led him to conclude that the sun was 18-20 times larger than the moon. It's not clear to me whether he had a size for the moon or whether he stopped at the relative size.

Quote

Aristarchus also reasoned that as the angular size of the Sun and the Moon were the same, but the distance to the Sun was between 18 and 20 times further than the Moon, the Sun must therefore be 18–20 times larger.

Question for math friends:

1. Given any, one planet in the observable universe, what are the chances that this planet has both (1) exactly one moon, which is also (2) the exact same apparent size as that planet's parent star?

Another question along those lines:

2. Under what conditions could ancient humans have measured the sun without the moon?

I'm ultimately getting to this thought experiment:

3. If you're a humanoid alien on an Andromedan planet with no moon, once a sage from your species apprehends the relationships between local triangles, what observations need to be made (or what technologies need to be invented) to allow the them to make accurate, celestial measurements?