Final answer:
Aristarchus calculated the size of the moon by observing a lunar eclipse and comparing the width of the Earth's shadow to the width of the Moon. Using this relationship, he estimated the size of the moon.
Step-by-step explanation:
Aristarchus calculated the size of the moon by observing a lunar eclipse. During a lunar eclipse, the Moon moves through the Earth's shadow, and Aristarchus observed that the shadow was about twice as wide as the Moon. He then assumed that the width of the shadow was also the width of the Earth, and by considering the relationship between the diameter of the Moon and the Earth, he was able to estimate the size of the moon.