This experiment is 3,000 years old. If the Earth were flat, shadows would be the same length if two points are located on the same longitude (I know FE'rs don't like the term 'longitude', but I'm not sure what else they'd like me to use here to get the point across - two places that are located directly South of one another at any distance, perhaps?)
However, this is not the case. A shadow off an obelisk in Alexandria, Egypt during the summer solstice is consistently in a different position than the shadow off an obelisk in Syene, Egypt (located directly South of Alexandria). This is due to the degree of curvature in the Earth between Alexandria and Syene.
Not only does this experiment prove the Earth is round, it also gives us the tools to determine the size of the Earth - 2.9 thousand years before we went to Space for the first time.
"Syene and Alexandria lie, he says, under the same meridian circle. Since meridian circles are great circles in the universe, the circles of the Earth which lie under them are necessarily also great circles. Thus, of whatever size this method shows the circle on the earth passing through Syene and Alexandria to be, this will be the greater size of the great circle of the earth. Now Eratosthenes asserts, and it is the fact, that Syene lies under the summer tropic. Whenever, therefore, the sun, being in Cancer at the summer solstice, is exactly in the middle of the heaven, the gnomons (pointers) of sundials necessarily throw no shadows, the position of the sun above them being exactly vertical; and
it is said that this is true throughout a space three hundred stades in diameter. But in Alexandria, at the same hour, the pointers of sundials throw shadows, because Alexandria lies further to the north than Syene. The two cities lying under the same meridian great circle, if we draw an arc form the extremity of the shadow to the base of the pointer of the sundial in Alexandria, the arc will be a segment of a great circle in the (hemispherical) bowl of the sundial, since the bowl of the sundial lies under the great circle (of the meridian). If we now conceive straight lines produced from each of the pointers through the earth, they will meet at the center of the earth. Since then the sundial at Syene is vertically under the sun, if we conceive a straight line coming from the sun to the top of the pointer of the sundial, the line reaching from the sun to the center of the earth will be on straight line. If we now conceive another straight line drawn upwards from the extremity of the shadow of the pointer of the sundial in Alexandria, through the top of the pointer to the sun, this straight line and the aforesaid straight line will be parallel, since they are straight lines coming through from different parts of the sun to different parts of the earth. On these straight lines, therefore, which are parallel, there falls the straight line drawn from the center of the earth to the pointer at Alexandria and the straight line drawn from the extremity of its shadow to the sun through the point (the top) where it meets the pointer. Now on this latter angle stands the arc carried round from the extremity of the shadow of the pointer to its base, while on the angle at the center of the earth stands the arc reaching from Syene to Alexandria. But the arcs are similar, since they stand on equal angles. Whatever ratio, therefore, the arc in the bowl of the sundial has to its proper circle, the arc reaching from Syene to Alexandria has that ratio to its proper circle. But the arc in the bowl is found to be one-fiftieth of its proper circle.
Therefore the distance from Syene to Alexandria must necessarily be one-fiftieth part of the great circle of the earth. And the said distance is 5000 stades; therefore the complete great circle measures 250,000 stades. Such is Eratosthenes‘ method."
http://www.academia.edu/27928173/How_the_Ancient_Egyptians_had_Calculated_the_Earths_Circumference_between_3750-1500_BC_a_revision_of_the_method_used_by_Eratosthenes