# Geometric explanations of approximations of $\pi$

Does any fast modern algorithm for approximating $\pi$ have a geometric interpretation as $\int \sqrt{1 - x^2}$ does?

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The most common modern method for approximating $\pi$ are Machin-like formulas. They work by writing $\pi$ as an integer combination of arctangents of small rational numbers, and then computing each arctangent by its power series.