the plane require numerical quadrature.) "An ingenious argument via elementary functions" shows the mean Euclidean distance between two points in the disk to be , while direct integration in polar coordinates shows the mean squared distance to be . If we are given an arbitrary location at a distance from the center of the disk, it is also of interest to determine the average distance from points in the distribution to this location and the average square of such distances. The latter value can be computed directly as . Average distance to an arbitrary internal point To find we need to look separately at the cases in which the location is internal or external, i.e. in which , and we find that in both cases the result can only be expressed in terms of complete elliptic integrals. If we consider an internal location, our aim (looking at the diagram) is to compute the expected value of under a distribution whose density is for , integrating in polar coordinates centered on the fixed location for which the area of a cell is ; hence Here can be found in terms of and using the Law of cosines. The steps needed to evaluate the integral, together with several references, will be found in the paper by Lew et al.; the result is that where and are complete elliptic integrals of the first and second kinds. ; . Average distance to an arbitrary external point Turning to an external location, we can set up the integral in a similar way, this time obtaining where the law of cosines tells us that and are the roots for of the equation Hence We may substitute to get using standard integrals. Hence again , while also See also Unit disk, a disk with radius one Annulus (mathematics), the region between two concentric circles Ball (mathematics), the usual term for the 3-dimensional analogue of a disk Disk algebra, a space of functions on a disk Disk segment Orthocentroidal disk, containing certain centers of a triangle References Euclidean geometry Circles Planar surfaces