You left out the most important part! The fact that in some places the sun is directly overhead is how we learned how big the Earth is.
The Greek-ish scholar Eratosthenes, who worked in the Library of Alexandria, once found a book which described that in one small town in the south of Egypt on a certain day of the year, the sun would be directly overhead. In Alexandria this never happens. He hired someone to go and pace the distance from this town to Alexandria and used the distance as well as the difference in angles between the sun in the town and the sun in Alexandria at noon to calculate the size of the earth to within a few percent.
