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A sphere (from Greek σφαίρα—sphaira, "globe, ball,"[1]) is a perfectly round geometrical object in three-dimensional space, such as the shape of a round ball. Like a circle in two dimensions, a perfect sphere is completely symmetrical around its center, with all points on the surface lying the same distance r from the center point. This distance r is known as the radius of the sphere. The maximum straight distance through the sphere is known as the diameter of the sphere. It passes through the center and is thus twice the radius.

In higher mathematics, a careful distinction is made between the surface of a sphere (referred to as a “sphere”), and the inside of a sphere (referred to as a “ball”). Thus, a sphere in three dimensions is considered to be a two-dimensional spherical surface embedded in three-dimensional Euclidean space, while a ball is a solid figure bounded by a sphere.


In their book Geometry and the Imagination[4] David Hilbert and Stephan Cohn-Vossen describe eleven properties of the sphere and discuss whether these properties uniquely determine the sphere. Several properties hold for the plane which can be thought of as a sphere with infinite radius. These properties are:

This property defines the sphere uniquely.
3. The sphere has constant width and constant girth.
4. All points of a sphere are umbilics.


5.The sphere does not have a surface of centers.
For a given normal section there is a circle whose curvature is the same as the sectional curvature, is tangent to the surface and whose center lines along on the normal line. Take the two centers corresponding to the maximum and minimum sectional curvatures: these are called the focal points, and the set of all such centers forms the focal surface.


6. All geodesics of the sphere are closed curves.
These properties define the sphere uniquely. These properties can be seen by observing soap bubbles. A soap bubble will enclose a fixed volume and due to surface tension it will try to minimize its surface area. This is why a free floating soap bubble approximates a sphere (though external forces such as gravity will distort the bubble's shape slightly).


9. The sphere has constant positive mean curvature.
10. The sphere has constant positive Gaussian curvature.


  1. Sphaira, Henry George Liddell, Robert Scott, A Greek-English Lexicon, at Perseus
  3. New Scientist | Technology | Roundest objects in the world created



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