Purely geometric derivation of the circle surface

Question

There are ways to get the surface or circumference of the circle by considering the areas and lengths of inscribed or Inscribing polygons. However this relies on using trig functions.

This is ok at first but ultimately trig functions when unpacked rely on coordinate definitions. If one could define trig functions without introducing any kind of coordinate system or basis that would be fine too.

Purely geometric in this case means completely clear of any coordinates or any kind of ortho normal basis.

Essentially what it means to geometrically calculate the surface of a circle is to geometrically exactly calculate pi, that is given some infinite series that converges to pi,or some recursive relation, show on purely geometric bases that that series in fact calculates the surface or circumference.

Archimedes used exhaustion to get an approximate value. Is there a way to get an exact value using similar axioms but no coordinates in any way.

Is this even possible, is it provably impossible?

Answer 1

Archimedes' Work

I believe that Archimedes did what the question is asking for. While it's true that he famously put bounds on $\pi$, he also

1. showed that the area of a circle should be exactly $\pi r^2$ (phrased in terms of a right triangle with the same area as the circle) by demonstrating that errors in polygonal approximations approach zero
2. had a method for obtaining approximations to pi that could be continued in a recursive way (though it would require better and better approximations of square roots along the way).

The first point is summarized here on Wikipedia and elaborated on in these notes from Bill Casselman's History of Mathematics course.

The second point is discussed at these notes for Chuck Lindsey's History of Mathematics course, which provide a translation of Archimedes' work for his approximation (how he did the trig stuff without the language of trig functions, essentially) and provide a recursive procedure that could be continued assuming arbitrary precision square roots.

Explicit Recurrence

While algebraic formulas did not exist in the time of Archimedes, an algebraic recurrence based on inscribed polygons and the Pythagoran Theorem is discussed in places such as this page of Nick Craig-Wood. At that page, they mention that a square inscribed in a circle of radius $1$ has side length $\sqrt{2}$ by the Pythagorean Theorem. They also mention that using the Pythagoran Theorem twice gives the following recurrence for the side length $d_n$ of an inscribed regular polygon with $n$ sides: $$d_{2n}=\sqrt{2-2\sqrt{1-\dfrac{d_{n}^{2}}{4}}}$$

We can turn this into a recurrence for a sequence of approximations of $\pi$. Let $p_n$ be the approximation corresponding to a polygon with $n$ sides. Since the circumference of this circle is $2\pi$, we have $d_n=2p_n/n$. Then we have

$$p_{2n}*\dfrac{2}{2n}=\sqrt{2-2\sqrt{1-\dfrac{\left(2p_{n}/n\right)^{2}}{4}}}$$

which reduces algebraically to $$p_{2n}=\sqrt{2n^{2}-2n\sqrt{n^{2}-p_{n}^{2}}}\text{.}$$

Starting with $p_4=2\sqrt{2}$, the first $10$ approximations to $\pi$ given by this recurrence are approximately $2.82843, 3.06147, 3.12145, 3.13655, 3.14033, 3.14128, 3.14151, 3.14157, 3.14159$.