If the areas of the four blue shapes labelled A, B, C and D are one unit each, what is the combined area of all the blue shapes? In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for π {\displaystyle \pi } . In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge.

Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of π {\displaystyle \pi } .It was not until 1882 that Ferdinand von Lindemann succeeded in proving more strongly that π is a transcendental number, and by doing so also proved the impossibility of squaring the circle with compass and straightedge.

Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. There exist in the hyperbolic plane ( countably) infinitely many pairs of constructible circles and constructible regular quadrilaterals of equal area, which, however, are constructed simultaneously. In 1837, Pierre Wantzel showed that lengths that could be constructed with compass and straightedge had to be solutions of certain polynomial equations with rational coefficients.In Chinese mathematics, in the third century CE, Liu Hui found even more accurate approximations using a method similar to that of Archimedes, and in the fifth century Zu Chongzhi found π ≈ 355 / 113 ≈ 3. Methods to calculate the approximate area of a given circle, which can be thought of as a precursor problem to squaring the circle, were known already in many ancient cultures. color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} . Contemporaneously with Antiphon, Bryson of Heraclea argued that, since larger and smaller circles both exist, there must be a circle of equal area; this principle can be seen as a form of the modern intermediate value theorem.

The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.Since the techniques of calculus were unknown, it was generally presumed that a squaring should be done via geometric constructions, that is, by compass and straightedge. Although much more precise numerical approximations to π {\displaystyle \pi } were already known, Kochański's construction has the advantage of being quite simple.