Squaring the circle
Squaring the circle is a problem proposed by ancient geometers. It is the challenge of constructing a square with the same area as a given circle by using only a finite number of steps with compass and straightedge. It may be taken to ask whether specified axioms of Euclidean geometry concerning the existence of lines and circles entail the existence of such a square.
In 1882, the task was proven to be impossible, as a consequence of the Lindemann–Weierstrass theorem which proves that pi (π) is a transcendental, rather than an algebraic irrational number; that is, it is not the root of any polynomial with rational coefficients. It had been known for some decades before then that the construction would be impossible if π were transcendental, but π was not proven transcendental until 1882. Approximate squaring to any given non-perfect accuracy, in contrast, is possible in a finite number of steps, since there are rational numbers arbitrarily close to π.
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 to mean the same thing as squaring the circle, but it may also refer to approximate or numerical methods for finding the area of a circle.
Methods to approximate the area of a given circle with a square were known already to Babylonian mathematicians. The Egyptian Rhind papyrus of 1800 BC gives the area of a circle as (64/81) d 2, where d is the diameter of the circle, and pi approximated to 256/81, a number that appears in the older Moscow Mathematical Papyrus and used for volume approximations (i.e. hekat). Indian mathematicians also found an approximate method, though less accurate, documented in the Sulba Sutras.Archimedes showed that the value of pi lay between 3 + 1/7 (approximately 3.1429) and 3 + 10/71 (approximately 3.1408). See Numerical approximations of π for more on the history.
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