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 10⁻ⁿ 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 ${\displaystyle S^{2}(1)}$ is a model for the two-dimensional elliptic plane. It carries an intrinsic metric that arises by measuring geodesic length. The geodesic circles are the parallels in a geodesic coordinate system.

More precisely, fix a point ${\displaystyle \mathbf {z} \in S^{2}(1)}$ that we place at the zenith. Associated to that zenith is a geodesic polar coordinate system ${\displaystyle (\phi ,\theta )}$, ${\displaystyle 0\leq \phi \leq \pi }$, ${\displaystyle 0\leq \theta <2\pi }$, where z is the point ${\displaystyle \phi =0}$. In these coordinates, the geodesic distance from z to any other point ${\displaystyle \mathbf {x} \in S^{2}(1)}$ having coordinates ${\displaystyle (\phi ,\theta )}$ is the value of ${\displaystyle \phi }$ 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 ${\displaystyle \mathbb {R} ^{3}}$, the spherical circle of radius ${\displaystyle R\leq \pi }$ centered at z is the set of x in ${\displaystyle S^{2}(1)}$ such that ${\displaystyle \mathbf {x} \cdot \mathbf {z} =\cos R}$.

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 ${\displaystyle S^{2}(\rho )}$ has radius of curvature ${\displaystyle \rho }$, then the area of the disk of radius R is given by

${\displaystyle A=2\pi \rho ^{2}(1-\cos(R/\rho )).}$

Observe that, as an application of L'Hôpital's rule, this tends to the Euclidean area ${\displaystyle \mathrm{\pi <!--\; \pi \; -->}{R}^{2}Observe\; that,\; as\; an\; application\; ofL\text{'}Hôpital\text{'}s\; rule,\; this\; tends\; to\; the\; Euclidean\; area}$${\displaystyle \pi R^{2}}$ in the flat limit ${\displaystyle \rho \to \infty }$.

The hyperbolic case is similar, with the area of a disk of intrinsic radius R in the (constant curvature ${\displaystyle -1}$) hyperbolic plane given by

${\displaystyle A=2\mathrm{\pi <!--\; \pi \; -->}(1-<!--\; -\; -->\cosh <!--\; \; -->R}$

The hyperbolic case is similar, with the area of a disk of intrinsic radius R in the (constant curvature ${\displaystyle -1}$) hyperbolic plane given by

where cosh is the hyperbolic cosine. More generally, for the constant curvature ${\displaystyle -k}$ hyperbolic plane, the answer is

${\displaystyle A=2\pi k^{-2}(1-\cosh(kR)).}$R in a flat space is always greater than the area of a spherical circle and smaller than a hyperbolic circle, provided all three circles have the same (intrinsic) radius. That is,

${\displaystyle 2\pi (1-\cos R)<\pi R^{2}<2\pi (1-\cosh R)}$${\displaystyle R>0}$. Intuitively, this is because the sphere tends to curve back on itself, yielding circles of smaller area than those in the plane, whilst the hyperbolic plane, when immersed into space, develops fringes that produce additional area. It is more generally true that the area of the circle of a fixed radius R is a strictly decreasing function of the curvature.

In all cases, if ${\displaystyle k}$ is the curvature (constant, positive or negative), then the isoperimetric inequality for a domain with area A and perimeter L is

${\displaystyle k}$