How to figure out how big the Earth is without too much effort

In ancient times, people were pretty clever.  One of these clever people was a greek by the name of Eratosthenes.  He noticed something unusual.  In his hometown of Alexandria Egypt, he noticed that the noon sun on the day of the Summer solstice was nearly overhead, but not quite.  Directly to the south was the town of Syene.  On the Summer solstice, the Sun was directly overhead - pretty much exactly at the zenith at noon.  Eratosthenes, being a very resourceful person, determined that the reason the Sun's location was different in the two places was because the Earth was curved.  This is illustrated below.
 
The location of the Sun as seen from the two locations in Egypt.  At Syene, the Sun is at the zenith at noon, while at Alexandria it is not exactly overhead.

While this is sort of neat, what use is it?  Well, Eratosthenes first determined how much difference there was in the Sun's location as viewed from the two locations.  Since the Sun was at the zenith at Syene, that means it was 90 degrees above the ground.  At Alexandria it was only 82.8 degrees above the ground - this gives and angle difference of 7.2 degrees in the Sun's location for the two cities.  So what good is that?  Well one of the neat things about geometry is how some angles can pop up over and over under the right circumstances.  Just take a look at the image below.  The difference in the zenith and the Sun's direction at Alexandria is 7.2 degrees.  This is also the difference between the latitudes of Syene and Alexandria.
 
The angles that show how the Sun's location varies and the latitude difference of the two locations is actually the same value!

Since there are 360 degrees in a circle, 7.2 degrees is 1/50 of an entire circle.  So the span between Syene and Alexandria corresponds to 1/50 of the way all the way around the Earth.  There are about 800 km between Syene and Alexandria, so that means there are 50 x 800 = 40,000 km all the way around the Earth.  This number is very close to the actual circumference of the Earth, 40,100 km.  So by just relating the angular difference to the difference in km, you can measure the size of the Earth pretty easily.

For those of you who are more mathematically minded, it is just a bunch of ratios. The ratio of the angle difference between the two locations compared to the entire number of degrees all the way around the Earth (360) has to equal the ratio of the distance between the two places and the entire circumference of the Earth. This can be mathematically written as

So if you know 3 of these things, you can always find the missing value.  It will take a little algebra and rearranging of things, but that isn't too bad.