of even-dimensional -balls of radius can be expressed in closed form: For the odd-dimensional analogue, where is the error function. Examples The 0-ball consists of a single point. The 0-dimensional Hausdorff measure is the number of points in a set. So, The 0-sphere consists of its two end-points, . So, The unit 1-ball is the interval of length 2. So, The unit 1-sphere is the unit circle in the Euclidean plane, and this has circumference (1-dimensional measure) The region enclosed by the unit 1-sphere is the 2-ball, or unit disc, and this has area (2-dimensional measure) Analogously, in 3-dimensional Euclidean space, the surface area (2-dimensional measure) of the unit 2-sphere is given by and the volume enclosed is the volume (3-dimensional measure) of the unit 3-ball, given by Recurrences The surface area, or properly the -dimensional volume, of the -sphere at the boundary of the -ball of radius is related to the volume of the ball by the differential equation or, equivalently, representing the unit -ball as a union of concentric -sphere shells, So, We can also represent the unit -sphere as a union of products of a circle (1-sphere) with an -sphere. Let and , so that and . Then, Since , the equation holds for all . This completes the derivation of the recurrences: Closed forms Combining the recurrences, we see that So it is simple to show by induction on that, where denotes the double factorial, defined for odd natural numbers by and similarly for even numbers . In general, the volume, in -dimensional Euclidean space, of the unit -ball, is given by where is the gamma function, which satisfies , , and , and so , and where we conversely define x! = for every x. By multiplying by , differentiating with respect to , and then setting , we get the closed form for the -dimensional surface of the sphere . Other relations The recurrences can be combined to give a "reverse-direction" recurrence relation for surface area, as depicted in the diagram: Index-shifting to then yields the recurrence relations: where , , and . The recurrence relation for can also be proved via integration with 2-dimensional polar coordinates: Spherical coordinates We may define a coordinate system in an -dimensional Euclidean space which is analogous to the spherical coordinate system defined for 3-dimensional Euclidean space, in which the coordinates consist of a radial coordinate , and