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.

Squaring the circle is a problem in geometry first proposed in Greek mathematics. 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. 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. 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 π ≈ 25 8 = 3.125 {\displaystyle \pi \approx {\tfrac {25}{8}}=3.125} , and at approximately the same time the ancient Egyptian mathematicians used π ≈ 256 81 ≈ 3.16 {\displaystyle \pi \approx {\tfrac {256}{81}}\approx 3.16} . Over 1000 years later, the Old Testament Books of Kings used the simpler approximation π ≈ 3 {\displaystyle \pi \approx 3} . [2] Ancient Indian mathematics, as recorded in the Shatapatha Brahmana and Shulba Sutras, used several different approximations to π {\displaystyle \pi } . [3] Archimedes proved a formula for the area of a circle, according to which 3 10 71 ≈ 3.141 < π < 3 1 7 ≈ 3.143 {\displaystyle 3\,{\tfrac {10}{71}}\approx 3.141<\pi <3\,{\tfrac {1}{7}}\approx 3.143} . [2] 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 π ≈ 355 / 113 ≈ 3.141593 {\displaystyle \pi \approx 355/113\approx 3.141593} , an approximation known as Milü. [4]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. [16] [17] In algebra, “to square” is to multiply a number by itself. For example, the square of 5 is 25 because 5 multiplied by 5 equals 25.

displaystyle \left(9 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. 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. displaystyle {\frac {6}{5}}\cdot \left(1+\varphi \right)=3.141\;{\color {red}640\;\ldots },} where φ {\displaystyle \varphi } is the golden ratio, φ = ( 1 + 5 ) / 2 {\displaystyle \varphi =(1+{\sqrt {5}})/2} .

If you have trouble locating the squared sign, you can search for it by typing “superscript two” in the “Search by keyword” field on the right. Simplify Radical Expressions Calculator to simplify radicals instead of finding fractional (decimal) answers.

The procedure for converting square inches to square feet or from acres to sq ft is the same as converting from square meters to square feet. In the following examples, you will find the most common of these conversions: how many square feet are in an acre. a 2 = x and a is an integer. For example, 4, 9 and 16 are perfect squares since their square roots, 2, 3 and 4, respectively, are integers.

x has a unique nonnegative square root r; this is called the principal square root .......... For example, the principal square root of 9 is sqrt(9) = +3, while the other square root of 9 is -sqrt(9) = -3. In common usage, unless otherwise specified, "the" square root is generally taken to mean the principal square root."[1]. 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. [1] Technically, it is not correct to talk about the square footage formula; we should rather talk about area formula. However, as long as we understand that a square footage formula is just another way to say that we will take an area formula and apply it using square feet, I think we can all agree the precise technical term doesn't matter much here. 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. [26] 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. [27] [28] Approximate constructions [ edit ]