<P> One method of deriving this formula, which originated with Archimedes, involves viewing the circle as the limit of a sequence of regular polygons . The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and the corresponding formula (that the area is half the perimeter times the radius, i.e. ⁄ × 2πr × r) holds in the limit for a circle . </P> <P> Although often referred to as the area of a circle in informal contexts, strictly speaking the term disk refers to the interior of the circle, while circle is reserved for the boundary only, which is a curve and covers no area itself . Therefore, the area of a disk is the more precise phrase for the area enclosed by a circle . </P> <P> Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis . However the area of a disk was studied by the Ancient Greeks . Eudoxus of Cnidus in the fifth century B.C. had found that the area of a disk is proportional to its radius squared . Archimedes used the tools of Euclidean geometry to show that the area inside a circle is equal to that of a right triangle whose base has the length of the circle's circumference and whose height equals the circle's radius in his book Measurement of a Circle . The circumference is 2πr, and the area of a triangle is half the base times the height, yielding the area π r for the disk . Prior to Archimedes, Hippocrates of Chios was the first to show that the area of a disk is proportional to the square of its diameter, as part of his quadrature of the lune of Hippocrates, but did not identify the constant of proportionality . </P> <P> A variety of arguments have been advanced historically to establish the equation A = π r 2 (\ displaystyle A = \ pi r ^ (2)) of varying degrees of mathematical rigor . The most famous of these is Archimedes' method of exhaustion, one of the earliest uses of the mathematical concept of a limit, as well as the origin of Archimedes' axiom which remains part of the standard analytical treatment of the real number system . The original proof of Archimedes is not quite rigorous by modern standards, because it assumes that we can compare the length of arc of a circle to the length of a secant and a tangent line, and similar statements about the area, as geometrically evident . </P>

Where did the formula for area of a circle come from