In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. However, they have a different character than squaring the circle, in that their solution involves the root of a cubic equation, rather than being transcendental. It takes only elementary geometry to convert any given rational approximation of π {\displaystyle \pi } into a corresponding compass and straightedge construction, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. It had been known for decades that the construction would be impossible if π {\displaystyle \pi } were transcendental, but that fact was not proven until 1882.

Therefore, more powerful methods than compass and straightedge constructions, such as neusis construction or mathematical paper folding, can be used to construct solutions to these problems. Bending the rules by introducing a supplemental tool, allowing an infinite number of compass-and-straightedge operations or by performing the operations in certain non-Euclidean geometries makes squaring the circle possible in some sense.James Gregory attempted a proof of the impossibility of squaring the circle in Vera Circuli et Hyperbolae Quadratura (The True Squaring of the Circle and of the Hyperbola) in 1667. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found. After the exact problem was proven unsolvable, some mathematicians applied their ingenuity to finding approximations to squaring the circle that are particularly simple among other imaginable constructions that give similar precision.

It is the challenge of constructing a square with the area of a given circle by using only a finite number of steps with a compass and straightedge. Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. 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.The more general goal of carrying out all geometric constructions using only a compass and straightedge has often been attributed to Oenopides, but the evidence for this is circumstantial. The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible. 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. displaystyle \left(9