Two other classical problems of antiquity, famed for their impossibility, were doubling the cube and trisecting the angle.

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. In 1882, it was proven that this figure cannot be constructed in a finite number of steps with an idealized compass and straightedge. In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem, which proves that pi ( π {\displaystyle \pi } ) is a transcendental number.

There is no method for starting with an arbitrary regular quadrilateral and constructing the circle of equal area. Although squaring the circle exactly with compass and straightedge is impossible, approximations to squaring the circle can be given by constructing lengths close to π {\displaystyle \pi } . To support this aim, members of the NRICH team work in a wide range of capacities, including providing professional development for teachers wishing to embed rich mathematical tasks into everyday classroom practice. The difficulty of the problem raised the question of whether specified axioms of Euclidean geometry concerning the existence of lines and circles implied the existence of such a square. 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.

Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle. color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} . 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 π {\displaystyle \pi } in the 5th decimal place. The solution of the problem of squaring the circle by compass and straightedge requires the construction of the number π {\displaystyle {\sqrt {\pi }}} , the length of the side of a square whose area equals that of a unit circle. 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.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 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.

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? 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). 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. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. This identity immediately shows that π {\displaystyle \pi } is an irrational number, because a rational power of a transcendental number remains transcendental.

In contrast, Eudemus argued that magnitudes cannot be divided up without limit, so the area of the circle would never be used up. If π {\displaystyle {\sqrt {\pi }}} were a constructible number, it would follow from standard compass and straightedge constructions that π {\displaystyle \pi } would also be constructible.

As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps.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. 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.