Eratosthenes used the curvature of the Earth to calculate its size. That was about 2200 years ago. The method he used is known and can be very easily and very cheaply checked nowadays by anyone who cares to do so. As a side effect, it also proves that the Earth isn't flat. That wasn't the point, of course. People back then already knew the world wasn't flat. There's an older surviving ancient Greek formal proof that the Earth is a sphere(*) and there may well have been even earlier proofs that haven't survived.
If the Earth is flat, I want a container made of sapient pearwood and I want to have a few beers with Nanny Ogg.
* Strictly speaking they were wrong - Earth isn't the perfect sphere they thought it was. They lacked the technology necessary to make accurate enough measurements to detect the slight deviation from a perfect sphere. But the relevant point is that it's very far from flat.
EDIT: In case anyone is curious enough to want to know the details but not curious enough to look for and read them, here's a very quick summary. He compared the shadows cast by two sticks of the same length at the same time in two different places with the same longitude. The difference between the shadows was due to the curvature of the Earth resulting in sunlight hitting the surface at different angles in the two places, so he could use that to calculate the angles of a triangle between the centre of the Earth and the two sticks and thus calculate what proportion of the circumference of the Earth lay between the two sticks. Since they were a known distance apart, simple multiplication then gives the circumference of the Earth.
