2-sphere

A sphere (from Greek — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball, (viz., analogous to a circular object in two dimensions).

Like a circle, which geometrically is an object in two-dimensional space, a sphere is defined mathematically as the set of points that are all at the same distance r from a given point, but in three-dimensional space. This distance r is the radius of the ball, and the given point is the center of the mathematical ball. The longest straight line through the ball, connecting two points of the sphere, passes through the center and its length is thus twice the radius; it is a diameter of the ball.

While outside mathematics the terms "sphere" and "ball" are sometimes used interchangeably, in mathematics a distinction is made between the sphere (a two-dimensional closed surface embedded in three-dimensional Euclidean space) and the ball (a three-dimensional shape that includes the sphere as well as everything inside the sphere). The ball and the sphere share the same radius, diameter, and center.

The surface area of a sphere is:

Archimedes first derived this formula from the fact that the projection to the lateral surface of a circumscribed cylinder (for example, the Lambert cylindrical equal-area projection) is area-preserving; it equals the derivative of the formula for the volume with respect to r because the total volume inside a sphere of radius r can be thought of as the summation of the surface area of an infinite number of spherical shells of infinitesimal thickness concentrically stacked inside one another from radius 0 to radius r. At infinitesimal thickness the discrepancy between the inner and outer surface area of any given shell is infinitesimal, and the elemental volume at radius r is simply the product of the surface area at radius r and the infinitesimal thickness.

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