Eratosthenes was a liberian in Alexandria. One day he stumbled upon a curious piece where he read that you could see the bottom of a deep well in Syene at noon on the day of the summer solstice. Alexandria was only 787km away and no such phenomenon occured there. He could tell that the sun was right above the well in Syene at noon whereas Alexendria recieve it at an angle.
He noticed that if he could calculate the angle the sun hits Alexandria with, he could use trigonometry to estimate the circumfrance of Earth. He placed a stuck a stick on the ground and measured the lenght of the shadow. He plugged in the lenght of the stick and shadow in the Law of Sines and found the angle the sun shone with over Alexandria.
Since a perpendicularly placed stick would have to pass through the center of the Earth if extended, he could use Alternate Interior Angles to work out the angle the distance between the two cities would correspond to. The graph below will brush up the high school remains of geometry knowledge you have.
Finally, he could proportion the angle to the distance to calculate the circumference:
7.2 / 360 = 787 / X
X = 39350 km
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