For some reason, when people ask about how π can be calculated (which is, to be fair, a nontrivial question and one worth thinking about), the Gregory-Leibniz series is a very popular response.
Stop doing that.
The Gregory-Leibniz series is itself a simple (but nontrivial) result that requires some basic calculus to establish, and therefore anyone asking about how to calculate π either: knows calculus and therefore also this series (and the more useful arcsine series); or doesn’t know calculus and thus can’t understand the relation to the arctangent. This is why it’s a horrible way to introduce laypeople to calculating π: it doesn’t appeal to any sort of geometric intuition at all.
Oh, and it converges sublinearly, which means that it cannot even be used for calculating π with reasonable speed. There are many tricks for series acceleration, but they generally all seem like black magic to people unfamiliar with them.
There’re ways of calculating π that appeal more closely to geometric intuition. Like, you know, finding the area of a quarter circle via numerical integration: . This actually also has not-so-nice convergence properties because of the nondifferentiability at one of the endpoints, unfortunately, but that is not difficult to overcome with some geometry.
Instead of integrating from 0 to 1, we could integrate from 0 to 1/√2. Geometrically, this corresponds to an eighth of a circle plus a triangle with base and height both 1/√2, and since is differentiable within that interval (infinitely differentiable even), the convergence might be expected to be less bad.
Or, similarly, integrating from 0 to 1/2, which gives a twelfth of a circle and a triangle with base 1/2 and height √3/2:
These both have the expected error when using the midpoint method or trapezoidal technique, in contrast to error when integrating over [0,1] using either rule.
Of course, these still have sublinear convergence and suck for actual computation, but they suck less and can actually be derived without any prior knowledge of calculus. (I sorta cheated by knowing that numerical integration has worse performance on nondifferentiable functions, but let’s overlook that— error is still better than .)
Disregarding all the above, the classical method of finding the areas of inscribing and circumscribing regular polygons with edge doubling does have linear convergence, so in fact Archimedes was way ahead of his time, and it’s telling that it was the method of calculating π for over a millennium. This does involve a square root extraction at every iteration, but that’s a small price to pay for converging at 2 bits per doubling. The trick of using where are respectively the areas of the circumscribing and inscribing polygons doubles the rate of convergence to 4 bits per iteration without increasing complexity, and has much the same motivation as Simpson’s rule. Armed with the knowledge of Taylor series, the faster convergence is easy to prove, but that defeats the purpose of this post!