section_path: Role and characterizations in mathematics/Geometry and trigonometry
section_name: Geometry and trigonometry
paragraph_text: Geometry and trigonometry
  
  π appears in formulae for areas and volumes of geometrical shapes based on circles, such as ellipses, spheres, cones, and tori. Below are some of the more common formulae that involve π.
  
  Some of the formulae above are special cases of the volume of the n -dimensional ball and the surface area of its boundary, the (n −1)-dimensional sphere, given below.
  
  Apart from circles, there are other curves of constant width. By Barbier's theorem, every curve of constant width has perimeter π 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 smooth and even algebraic curves of constant width.
  
  Definite integrals that describe circumference, area, or volume of shapes generated by circles typically have values that involve π. For example, an integral that specifies half the area of a circle of radius one is given by: ∫ − 1 1 1 − x 2 d x = π 2. {\displaystyle \int _{-1}^{1}{\sqrt {1-x^{2}}}\,dx={\frac {\pi }{2}}.}
  
  In that integral, the function 1 − x 2 {\displaystyle {\sqrt {1-x^{2}}}} represents the height over the x {\displaystyle x} -axis of a semicircle (the square root is a consequence of the Pythagorean theorem), and the integral computes the area below the semicircle.