It was calculated by very, very intelligent mathematicians using ancient instruments.
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Electro-Mechanical Computers were used before first generation of computers.
No, the value of pi was not first calculated by Budhayana. It is debated who the first person was to calculate it. However, it is believed that Archimedes was the first to calculate it using polygons, while Ptolemy was the first to assigned it its current value directly.
before electronic computers there were human computers (i.e. humans doing calculations with pencil and paper, if they were lucky a mechanical desk calculator and a book of precalculated mathematical tables).
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Archimedes made a variety of advances including volume and buoyancy measurements along with mathematics that define the size of a cylinder using a sphere. He also helped to make a lever more useful in farming.