radius and the infinitesimal thickness. At any given radius , the incremental volume () equals the product of the surface area at radius () and the thickness of a shell (): The total volume is the summation of all shell volumes: In the limit as approaches zero this equation becomes: Substitute : Differentiating both sides of this equation with respect to yields as a function of : This is generally abbreviated as: where is now considered to be the fixed radius of the sphere. Alternatively, the area element on the sphere is given in spherical coordinates by . In Cartesian coordinates, the area element is The total area can thus be obtained by integration: The sphere has the smallest surface area of all surfaces that enclose a given volume, and it encloses the largest volume among all closed surfaces with a given surface area. The sphere therefore appears in nature: for example, bubbles and small water drops are roughly spherical because the surface tension locally minimizes surface area. The surface area relative to the mass of a ball is called the specific surface area and can be expressed from the above stated equations as where is the density (the ratio of mass to volume). Other geometric properties A sphere can be constructed as the surface formed by rotating a circle about any of its diameters; this is essentially the traditional definition of a sphere as given in Euclid's Elements. 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. A sphere is uniquely determined by four points that are not coplanar. More generally, a sphere is uniquely determined by four conditions such as passing through a point, being tangent to a plane, etc. This property is analogous to the property that three non-collinear points determine a unique circle in a plane. Consequently, a sphere is uniquely determined by (that is, passes through) a circle and a point not in the plane of that circle. By examining the common solutions of the equations of two spheres, it can be seen that two spheres intersect in a circle and the plane containing that circle is called the radical plane of the intersecting spheres. Although the radical plane is