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The real reason that pi was never calculated at high precision is that for day to day applications, ancient cultures didn't need a high precision. Ancient equipment generally had a lower accuracy compared to modern tools. For instance, there is no need to know pi in six decimals, if you can only measure the radius to an accuracy of two decimals.

Even nowadays, there is no practical use for a million digits of pi, simply because we cannot measure anything in the physical world to that accuracy. For instance, one of the most precise measurements possible today relies on the Mössbauer Effect and will yield results accurate to 14 digits. That's an error of one second in 3 million years. Also note that 39 digits of pi suffice to calculate the circumference of the known universe from its radius to within the diameter of the hydrogen atom1.

1: Fractals, Chaos, Power Laws, Minutes from an Infinite Paradise, Manfred Schroeder, New York, W.H. Freeman and Company, 1991