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The following is a list of significant formulae involving the mathematical constant π. Many of these formulae can be found in the article Pi, or the article Approximations of π.
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 2 × 2πr × r, holds for a circle.
The Taylor series for the inverse tangent function, often called Gregory's series, is The Leibniz formula is the special case [3] It also is the Dirichlet L -series of the non-principal Dirichlet character of modulus 4 evaluated at and therefore the value β(1) of the Dirichlet beta function.
In mathematics, Euler's identity[note 1] (also known as Euler's equation) is the equality where e {\displaystyle e} is Euler's number, the base of natural logarithms, i {\displaystyle i} is the imaginary unit, which by definition satisfies i 2 = − 1 {\displaystyle i^ {2}=-1} , and π {\displaystyle \pi } is pi, the ratio of the circumference of a circle to its diameter. Euler's identity is ...
In engineering, applied mathematics, and physics, the Buckingham π theorem is a key theorem in dimensional analysis. It is a formalisation of Rayleigh's method of dimensional analysis. Loosely, the theorem states that if there is a physically meaningful equation involving a certain number n of physical variables, then the original equation can ...
The circumference of a circle is related to one of the most important mathematical constants. This constant, pi, is represented by the Greek letter The first few decimal digits of the numerical value of are 3.141592653589793 ... [4] Pi is defined as the ratio of a circle's circumference to its diameter Or, equivalently, as the ratio of the circumference to twice the radius. The above formula ...
C = 2 π r {\displaystyle C=2\pi r} is the circumference (the length of any one of its great circles), S {\displaystyle S} is the surface area, V {\displaystyle V} is the volume.
In the 3rd century BCE, Archimedes proved the sharp inequalities 223 ⁄ 71 < π < 22 ⁄ 7, by means of regular 96-gons (accuracies of 2·10 −4 and 4·10 −4, respectively). [ 15 ] In the 2nd century CE, Ptolemy used the value 377 ⁄ 120 , the first known approximation accurate to three decimal places (accuracy 2·10 −5 ). [ 16 ]