Area of a circle 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 2"r", and the area of a triangle is half the base times the height, yielding the area "r" for the disk. Prior to Archimedes, Hippocrates of Chios was the first
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