Area of a disk

The area of a disk, more commonly called the area of a circle, of radius r is equal to πr². Here the symbol π (Greek letter pi) denotes the constant ratio of the circumference of a circle to its diameter or of the area of a circle to the square of its radius. Since the area of a regular polygon is half its perimeter times its apothem, and a regular polygon approaches a circle as the number of sides increase, the area of a disk is half its circumference times its radius (i.e. ¹⁄₂ × 2πr × r).

History

Modern mathematics can obtain the area using the methods of integral calculus or its more sophisticated offspring, real analysis. However the area of circles was studied by the Ancient Greeks. Eudoxus of Cnidus in the fifth century B.C. had found that the areas of circles are proportional to their radius squared. The great mathematician 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 πr² 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.