I know that modern sciences have many many applications for the number PI, many of them outside of geometry, but I do not understand what practical applications had this constant in the ancient world.
What motivated the Greeks, Babylonians and the Egyptians to try to calculate this number?
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For all practical applications in ancient time daily life, the various rational approximations in vogue where more than enough, even considering that the instruments to measure the length where prone to higher errors.
I think the interest in finding ever more precise value of $\pi$ was actually due to its irrationality, as given by the method of exhaustion, and thus on the existence of incommensurable quantities, infinite convergent series, etc. And later, that it is transcendent, so the impossibility to solve the famous problem of Quadratura Circuli.
There is an interesting resume of the history in the world of pi
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If you are interested in the volume of a heap of wheat (which looks like a cone) or in the content of a granary (which may look like a cylinder), then you need approximations of $\pi$. The Babylonians used $\pi = 3$ in plane geometry, and the approximation $\pi = 3 \frac18$ only occurs in connection with computing the volume of solids.
Pi appears in equations for volumes of regular solids, as well as in the area of a circle, among many other locations in mathematics. Knowing how to find these geometric pieces of information was valuable to ancient civilizations due to the precision required in their many construction projects of scales varying from small to colossal. For example, knowing how much stone is needed to construct a pillar of certain dimensions requires knowledge of pi.