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Plato’s Approximation of Pi

Did Plato approximate π? There is no evidence that he did so. The myth that he did seems to have originated in the book The Open Society and its Enemies by the British philosopher Karl Popper (1902–1994) [1]. In note 9 to Chapter 6, Popper writes: “It is a curious fact that √ 2 + √3 very nearly approximates π . . . . A kind of explanation of this curious fact is that it follows from the fact that the arithmetical mean of the areas of the circumscribed hexagon and the inscribed octagon is a good approximation of the area of the circle.” After noting that Plato was interested in adding irrational numbers and must have added √ 2 + √3, Popper writes: “It seems a plausible hypothesis that Plato knew of [ √ 2 + √3 ≈ π], but was unable to prove whether or not is was a strict inequality or only an approximation. . . . I must again emphasize that no direct evidence is known to me to show that this was in Plato’s mind: but if we consider the indirect evidence here marshaled, then the hypothesis does perhaps not seem too far-fetched.” The approximation is actually quite good, as it yields π ≈ 3.146264 for a relative error of about 0.1487%. The purpose of this note is to establish the inequality in the approximation, that is, to show analytically that π < √ 2 + √3. We do this with simple rational approximations to the three constants. For the square roots we have √ 2 > 41/29 since 2 > 1681/841, and √ 3 > 71/41 since 3 > 5041/1681. Thus, √ 2 + √3 > 41 29 + 71 41 = 3 173 1189 > 3 1 7 > π.