Yes, he came up with 24,500 miles -- not bad!! He also determined that the Earth's oceans were interconnected and calculated that a ship launched westward across the Atlantic could reach India. This was about 1600 years before there's a record of Columbus trying it -- much too long for such a theory to go untested IMO. Check out the links at: http://ce.eng.usf.edu/pharos/Alexandria/links.html
As for the Sphinx's nose, the defacement occurred much earlier, probably as the result of a Muslim invasion around 1378 a.d. Napoleon's troops have been blamed for doing it with a cannon but that story appears to be untrue. Quite a few people had a crack at it. Defacement of statues of other cultures' deities has been a popular pastime throughout history.