I'm tempted to say that unless there is a trick here, there can't be an exact method because that would compromise pi's status.
It reminds me of reading a book about the ancient Egyptians. It said that they used a measuring device consisting of a wheel on a stick to mark out the site when planning a pyramid. But they assumed that the circumference was exactly 3 times the diameter, so built in an error into their pyramid proportions, because the sides were slightly shorter than they thought.
So, the book claimed, that explained the consistent but unusual angle of all the earlier pyramids. Later ones were slightly less steep, after they corrected their mistake.
Nice story, but might well be untrue.
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A less mildly diverting brain teaser new