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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. 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.
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? 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. Approximate constructions with any given non-perfect accuracy exist, and many such constructions have been found.
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.
Despite the proof that it is impossible, attempts to square the circle have been common in pseudomathematics (i. 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. 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 was not until 1882 that Ferdinand von Lindemann proved the transcendence of π {\displaystyle \pi } and so showed the impossibility of this construction.
As well, several later mathematicians including Srinivasa Ramanujan developed compass and straightedge constructions that approximate the problem accurately in few steps. Although the circle cannot be squared in Euclidean space, it sometimes can be in hyperbolic geometry under suitable interpretations of the terms. Hippocrates of Chios attacked the problem by finding a shape bounded by circular arcs, the lune of Hippocrates, that could be squared. 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 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. 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. 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. This identity immediately shows that π {\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 e {\displaystyle e} , to show that π {\displaystyle \pi } is transcendental and therefore that squaring the circle is impossible.Having taken their lead from this problem, I believe, the ancients also sought the quadrature of the circle.
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