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In geometry, the area enclosed by a circle of radius r is πr2. Here the Greek letter π represents the constant ratio of the circumference of any circle to its diameter, approximately equal to 3.14159.

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–namely, A = 1/2 × 2πr × r, holds in the limit for a circle.

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.

Unit circle area Monte Carlo integration. Estimate by these 900 samples is 4×709/900 = 3.15111...

When more efficient methods of finding areas are not available,

When more efficient methods of finding areas are not available, we can resort to “throwing darts”. This Monte Carlo method uses the fact that if random samples are taken uniformly scattered across the surface of a square in which a disk resides, the proportion of samples that hit the disk approximates the ratio of the area of the disk to the area of the square. This should be considered a method of last resort for computing the area of a disk (or any shape), as it requires an enormous number of samples to get useful accuracy; an estimate good to 10n requires about 100n random samples (Thijssen 2006, p. 273).

Finite rearrangement

We have seen that by partitioning the disk into an infinite number of pieces we can reassemble the pieces into a rectangle. A remarkable fact discovered relatively recently (Laczkovich 1990) is that we can dissect the disk into a large but finite number of pieces and then reassemble the pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence of such a partition (in fact, of many such partitions) but does not exhibit any particular partition.

Non-Euclidean circles

Circles can be defined in non-Euclidean geometry, and in particular in the hyperbolic and We have seen that by partitioning the disk into an infinite number of pieces we can reassemble the pieces into a rectangle. A remarkable fact discovered relatively recently (Laczkovich 1990) is that we can dissect the disk into a large but finite number of pieces and then reassemble the pieces into a square of equal area. This is called Tarski's circle-squaring problem. The nature of Laczkovich's proof is such that it proves the existence of such a partition (in fact, of many such partitions) but does not exhibit any particular partition.

Non-Euclidean circlesCircles can be defined in non-Euclidean geometry, and in particular in the hyperbolic and elliptic planes.

For example, the unit sphere

More precisely, fix a point that we place at the zenith. Associated to that zenith is a geodesic polar coordinate system , , , where z is the point . In these coordinates, the geodesic distance from z to any other point having coordinates is the value of at x. A spherical circle is the set of points a geodesic distance R from the zenith point z. Equivalently, with a fixed embedding into , the spherical circle of radius centered at z is the set of x in such that .

We can also measure the area of the spherical disk enclosed within a spherical circle, using the intrinsic surface area measure on the sphere. The area of the disk of radius R is then given by

More generally, if a sphere has radius of curvature , then the area of the disk of radius R is given by

Observe that, as an application of L'Hôpital's rule, this tends to the Euclidean area

The hyperbolic case is similar, with the area of a disk of intrinsic radius R in the (constant curvature ) hyperbolic plane given by