The points on the sphere with radius formula_12 and center formula_13 can be parameterized via The parameter formula_15 can be associated with the angle counted positive from the direction of the positive "z"-axis through the center to the radius-vector, and the parameter formula_16 can be associated with the angle counted positive from the direction of the positive "x"-axis through the center to the projection of the radius-vector on the "xy"-plane. A sphere of any radius centered at zero is an integral surface of the following differential form:  This equation reflects that position and velocity vectors of a point, and , traveling on the sphere are always orthogonal to each other. A sphere can also be constructed as the surface formed by rotating a circle about any of its diameters. Since a circle is a special type of ellipse, a sphere is a special type of ellipsoid of revolution. Replacing the circle with an ellipse rotated about its major axis, the shape becomes a prolate spheroid; rotated about the minor axis, an oblate spheroid. In three dimensions, the volume inside a sphere (that is, the volume of a ball, but classically referred to as the volume of a sphere) is where is the radius and is the diameter of the sphere. Archimedes first derived this formula by showing that the volume inside a sphere is twice the volume between the sphere and the circumscribed cylinder of that sphere (having the height and diameter equal to the diameter of the sphere). This may be proved by inscribing a cone upside down into semi-sphere, noting that the area of a cross section of the cone plus the area of a cross section of the sphere is the same as the area of the cross section of the circumscribing cylinder, and applying Cavalieri's principle. This formula can also be derived using integral calculus, i.e. disk integration to sum the volumes of an infinite number of circular disks of infinitesimally small thickness stacked side by side and centered along the -axis from to , assuming the sphere of radius is centered at the origin. At any given , the incremental volume () equals the product of the cross-sectional area of the disk at and its thickness ():  The total volume is the summation of all incremental volumes:  In the limit as approaches zero, this equation becomes:  At any given , a right-angled triangle connects , and to the origin; hence, applying the Pythagorean theorem yields:  Using this substitution gives  which can be evaluated to give the result  An alternative formula is found using spherical coordinates, with volume element so For most practical purposes, the volume inside a sphere inscribed in a cube can be approximated as 52.4% of the volume of the cube, since , where is the diameter of the sphere and also the length of a side of the cube and  ≈ 0.5236.