In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi (${\displaystyle \pi }$) is a transcendental number. That is, ${\displaystyle \pi }$ is not the root of any polynomial with rational coefficients. It had been known for decades that the construction would be impossible if ${\displaystyle \pi }$ were transcendental, but that fact was not proven until 1882. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.

Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i.e. the work of mathematical cranks). The expression "squaring the circle" is sometimes used as a metaphor for trying to do the impossible.

The term quadrature of the circle is sometimes used as a synonym for squaring the circle. It may also refer to approximate or numerical methods for finding the area of a circle. In general, quadrature or squaring may also be applied to other plane figures.

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. These methods can be summarized by stating the approximation to π that they produce. In around 2000 BCE, the Babylonian mathematicians used the approximation ${\displaystyle \pi \approx {\tfrac {25}{8}}=3.125}$, and at approximately the same time the ancient Egyptian mathematicians used ${\displaystyle \pi \approx {\tfrac {256}{81}}\approx 3.16}$. Over 1000 years later, the Old Testament Books of Kings used the simpler approximation ${\displaystyle \pi \approx 3}$. Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to ${\displaystyle \pi }$. Archimedes proved a formula for the area of a circle, according to which ${\displaystyle 3\,{\tfrac {10}{71}}\approx 3.141<\pi <3\,{\tfrac {1}{7}}\approx 3.143}$. 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 ${\displaystyle \pi \approx 355/113\approx 3.141593}$, an approximation known as Milü.

The problem of constructing a square whose area is exactly that of a circle, rather than an approximation to it, comes from Greek mathematics. Greek mathematicians found compass and straightedge constructions to convert any polygon into a square of equivalent area. They used this construction to compare areas of polygons geometrically, rather than by the numerical computation of area that would be more typical in modern mathematics. As Proclus wrote many centuries later, this motivated the search for methods that would allow comparisons with non-polygonal shapes:

The first known Greek to study the problem was Anaxagoras, who worked on it while in prison. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. Antiphon the Sophist believed that inscribing regular polygons within a circle and doubling the number of sides would eventually fill up the area of the circle (this is the method of exhaustion). Since any polygon can be squared, he argued, the circle can be squared. In contrast, Eudemus argued that magnitudes can be divided up without limit, so the area of the circle would never be used up. 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 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 problem of finding the area under an arbitrary curve, now known as integration in calculus, or quadrature in numerical analysis, was known as squaring before the invention of calculus. 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. For example, Newton wrote to Oldenburg in 1676 "I believe M. Leibnitz will not dislike the theorem towards the beginning of my letter pag. 4 for squaring curve lines geometrically". In modern mathematics the terms have diverged in meaning, with quadrature generally used when methods from calculus are allowed, while squaring the curve retains the idea of using only restricted geometric methods.

A 1647 attempt at squaring the circle, Opus geometricum quadraturae circuli et sectionum coni decem libris comprehensum by Grégoire de Saint-Vincent, was heavily criticized by Vincent Léotaud. Nevertheless, de Saint-Vincent succeeded in his quadrature of the hyperbola, and in doing so was one of the earliest to develop the natural logarithm. James Gregory, following de Saint-Vincent, attempted another 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. Although his proof was faulty, it was the first paper to attempt to solve the problem using algebraic properties of ${\displaystyle \pi }$. Johann Heinrich Lambert proved in 1761 that ${\displaystyle \pi }$ is an irrational number. 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.

After Lindemann's impossibility proof, the problem was considered to be settled by professional mathematicians, and its subsequent mathematical history is dominated by pseudomathematical attempts at circle-squaring constructions, largely by amateurs, and by the debunking of these efforts. As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps.

Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle. Like squaring the circle, these cannot be solved by 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. 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.

The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number ${\displaystyle {\sqrt {\pi }}}$, the length of the side of a square whose area equals that of a unit circle. If ${\displaystyle {\sqrt {\pi }}}$ were a constructible number, it would follow from standard compass and straightedge constructions that ${\displaystyle \pi }$ would also be constructible. 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. Thus, constructible lengths must be algebraic numbers. If the circle could be squared using only compass and straightedge, then ${\displaystyle \pi }$ would have to be an algebraic number. It was not until 1882 that Ferdinand von Lindemann proved the transcendence of ${\displaystyle \pi }$ and so showed the impossibility of this construction. Lindemann's idea was to combine the proof of transcendence of Euler's number ${\displaystyle e}$, shown by Charles Hermite in 1873, with Euler's identity $${\displaystyle e^{i\pi }=-1.}$$ This identity immediately shows that ${\displaystyle \pi }$ is an irrational number, because a rational power of a transcendental number remains transcendental. Lindemann was able to extend this argument, through the Lindemann–Weierstrass theorem on linear independence of algebraic powers of ${\displaystyle e}$, to show that ${\displaystyle \pi }$ is transcendental and therefore that squaring the circle is impossible.

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. For example, Dinostratus' theorem uses the quadratrix of Hippias to square the circle, meaning that if this curve is somehow already given, then a square and circle of equal areas can be constructed from it. The Archimedean spiral can be used for another similar construction. Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. The hyperbolic plane does not contain squares (quadrilaterals with four right angles and four equal sides), but instead it contains regular quadrilaterals, shapes with four equal sides and four equal angles sharper than right angles. 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. There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Symmetrically, there is no method for starting with an arbitrary circle and constructing a regular quadrilateral of equal area, and for sufficiently large circles no such quadrilateral exists.

Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to ${\displaystyle \pi }$. It takes only elementary geometry to convert any given rational approximation of ${\displaystyle \pi }$ into a corresponding compass and straightedge construction, but such constructions tend to be very long-winded in comparison to the accuracy they achieve. 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.

Kochański's approximate construction

Continuation with equal-area circle and square; ${\displaystyle r}$ denotes the initial radius

One of many early historical approximate compass-and-straightedge constructions is from a 1685 paper by Polish Jesuit Adam Adamandy Kochański, producing an approximation diverging from ${\displaystyle \pi }$ in the 5th decimal place. Although much more precise numerical approximations to ${\displaystyle \pi }$ were already known, Kochański's construction has the advantage of being quite simple. In the left diagram $${\displaystyle |P_{3}P_{9}|=|P_{1}P_{2}|{\sqrt {{\frac {40}{3}}-2{\sqrt {3}}}}\approx 3.141\,5{\color {red}33\,338}\cdot |P_{1}P_{2}|\approx \pi r.}$$ In the same work, Kochański also derived a sequence of increasingly accurate rational approximations for ${\displaystyle \pi }$.