Circle a triangle whose base has the length of the circle's circumference and whose height equals the circle's radius, which comes to multiplied by the radius squared: Equivalently, denoting diameter by "d", that is, approximately 79% of the circumscribing square (whose side is of length "d"). The circle is the plane curve enclosing the maximum area for a given arc length. This relates the circle to a problem in the calculus of variations, namely the isoperimetric inequality. Equation of a circle<br> In an "x"–"y" Cartesian coordinate system, the circle with centre coordinates ("a", "b") and radius "r" is the set of
Area of a circle formula_53. The calculations Archimedes used to approximate the area numerically were laborious, and he stopped with a polygon of 96 sides. A faster method uses ideas of Willebrord Snell ("Cyclometricus", 1621), further developed by Christiaan Huygens ("De Circuli Magnitudine Inventa", 1654), described in . Given a circle, let "u" be the perimeter of an inscribed regular "n-"gon, and let "U" be the perimeter of a circumscribed regular "n-"gon. Then "u" and "U" are lower and upper bounds for the circumference of the circle that become sharper and sharper as "n" increases, and their average ("u" + "U")/2 is an especially