Why did Aristarchus choose a half Moon to calculate the distance from Earth to the Sun?

Aristarchus chose a half Moon for his calculations because it allowed him to create a right triangle using the Earth, the Moon, and the Sun. At the phase of the half Moon, the angle between the line of sight to the Moon and the line of sight to the Sun forms a right angle at the Moon. This geometric configuration is essential for applying trigonometric principles to determine distances in space.

By understanding the relationship between the lengths of the sides of the right triangle and utilizing the known distance from the Earth to the Moon, Aristarchus was able to infer the distance from the Earth to the Sun. This method relies on the properties of right triangles, specifically the sine and cosine functions, which are fundamental in physics and astronomy for calculating distances and angles.

The other options do not facilitate the necessary geometric relationships needed for such calculations. The half Moon phase provides the ideal observational geometry to derive the distances effectively.