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 "area" and "circumference". The analytical definitions are seen to be equivalent, if it is agreed that the circumference of the circle is measured as a rectifiable curve by means of the integral The integral appearing on the right is an abelian integral whose value is a half-period of the sine function, equal to . Thus formula_6 is seen to be true as a theorem. Several of the arguments that follow use only concepts from elementary calculus to reproduce the formula formula_1, but in many cases to regard these as actual proofs, they rely implicitly on the fact that one can