Editing Pi (section)

=== Geometry and trigonometry ===
[[File:Circle Area.svg|thumb|alt=A diagram of a circle with a square coving the circle's upper right quadrant.|right|The area of the circle equals {{pi}} times the shaded area. The area of the [[unit circle]] is {{pi}}.]]

{{pi}} appears in formulae for areas and volumes of geometrical shapes based on circles, such as [[ellipse]]s, [[sphere]]s, [[cone (geometry)|cones]], and [[torus|tori]].<ref>{{mathworld| |title=Circle |id=Circle|ref=none}}</ref>  Below are some of the more common formulae that involve {{pi}}.<ref>{{harvnb|Bronshteĭn|Semendiaev|1971|pp=200, 209}}.</ref>
* The circumference of a circle with radius {{math|''r''}} is {{math|2π''r''}}.<ref>{{mathworld |title=Circumference |id=Circumference|ref=none}}</ref>
* The [[area of a disk|area of a circle]] with radius {{math|''r''}} is {{math|π''r''<sup>2</sup>}}.
* The area of an ellipse with semi-major axis {{math|''a''}} and semi-minor axis {{math|''b''}} is {{math|π''ab''}}.<ref>{{mathworld |title=Ellipse |id=Ellipse|ref=none}}</ref>
* The volume of a sphere with radius {{math|''r''}} is {{math|{{sfrac|4|3}}π''r''<sup>3</sup>}}.
* The surface area of a sphere with radius {{math|''r''}} is {{math|4π''r''<sup>2</sup>}}.
Some of the formulae above are special cases of the volume of the [[N-ball|''n''-dimensional ball]] and the surface area of its boundary, the [[n-sphere|(''n''−1)-dimensional sphere]], given [[#The gamma function and Stirling's approximation|below]].

Apart from circles, there are other [[Curve of constant width|curves of constant width]]. By [[Barbier's theorem]], every curve of constant width has perimeter {{pi}} times its width. The [[Reuleaux triangle]] (formed by the intersection of three circles with the sides of an [[equilateral triangle]] as their radii) has the smallest possible area for its width and the circle the largest. There also exist non-circular [[Smoothness|smooth]] and even [[algebraic curve]]s of constant width.<ref>{{cite book |last1=Martini |first1=Horst |last2=Montejano |first2=Luis |last3=Oliveros |first3=Déborah |author3-link=Déborah Oliveros |doi=10.1007/978-3-030-03868-7 |isbn=978-3-030-03866-3 |mr=3930585 |publisher=Birkhäuser |s2cid=127264210 |title=Bodies of Constant Width: An Introduction to Convex Geometry with Applications |year=2019}}{{pb}}
See Barbier's theorem, Corollary 5.1.1, p. 98; Reuleaux triangles, pp. 3, 10; smooth curves such as an analytic curve due to Rabinowitz, §&nbsp;5.3.3, pp. 111–112.</ref>

[[Integral|Definite integrals]] that describe circumference, area, or volume of shapes generated by circles typically have values that involve {{pi}}. For example, an integral that specifies half the area of a circle of radius one is given by:<ref>{{cite book |last1=Herman |first1=Edwin |last2=Strang |first2=Gilbert |author2-link=Gilbert Strang |contribution=Section 5.5, Exercise 316 |contribution-url=https://openstax.org/books/calculus-volume-1/pages/5-5-substitution |page=594 |publisher=[[OpenStax]] |title=Calculus |volume=1 |year=2016}}</ref>

<math display=block>\int_{-1}^1 \sqrt{1-x^2}\,dx = \frac{\pi}{2}.</math>

In that integral, the function <math>\sqrt{1-x^2}</math> represents the height over the <math>x</math>-axis of a [[semicircle]] (the [[square root]] is a consequence of the [[Pythagorean theorem]]), and the integral computes the area below the semicircle.  The existence of such integrals makes {{pi}} an [[Period (algebraic geometry)|algebraic period]].<ref>{{cite book |last1=Kontsevich |first1=Maxim |author-link1=Maxim Kontsevich |contribution=Periods |date=2001 |title=Mathematics Unlimited — 2001 and Beyond |pages=771–808 |editor-last=Engquist |editor-first=Björn |doi=10.1007/978-3-642-56478-9_39 |location=Berlin, Heidelberg |publisher=Springer |language=en |isbn=978-3-642-56478-9 |last2=Zagier |first2=Don |editor2-last=Schmid |editor2-first=Wilfried}}</ref>