But then his successor Aristarchus under took to measure the sun-earth distance as follows:
He looked at the angle the earth and moon form when the moon is exactly half full. This would give the sun distance as a multiple of the moon's distance using trigonometry.
He observed the earth's shadow during lunar eclipses. This gave the moon's size in terms of earth size.
He knew the earth's size from measuring shadows at different latitudes on earth and the distance between those sites.
Knowing the moons size vs apparent size, simple trigonometry gave him the distance to the moon. And therefor also the distance to the sun.
Unfortunately his angle measuring techniques were primitive and he decided the sun was 20x further from earth than the moon (rather than about 400x). But his other figures are quite passable.