一个半径为 r 的圆的面积为 πr。这里的希腊字母π，和通常一样代表圆周长和直径的比值，即圆周率。

现代数学家可以用微积分或更高深的后继理论实分析得到这个面积。但是，在古希腊伟大的数学家阿基米德在《圆的测量》（Measurement of a Circle）中使用欧几里得几何证明了一个圆周内部的面积等于一个以其圆周长及半径作为两个直角边的直角三角形面积。周长为 2πr，直角三角形的面积为两直角边乘积的一半，得出圆的面积为 πr。中国古代流传之《九章算术·方田》章中的圆田术对圆面积计算的叙述为“半周半径相乘得积步”。魏晋时代的刘徽注解《九章算术》时，则以“穷尽”割圆术提供了相同结果的证明。

In mathematics, the disk enclosed by a circle of radius r has area πr. Here the Greek letter π represents a constant, approximately equal to 3.14159, which is equal to the ratio of the circumference of any circle to its diameter.

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. ⁄₂ × 2πr × r) holds in the limit for a circle.