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sphere





sphere
Roughly speaking, a ball-shaped object. In everyday usage a sphere is often considered to be solid; mathematicians call this the interior of the sphere. In mathematics, a sphere is a quadric consisting only of a surface and is therefore hollow. More precisely, a sphere is the set of all points in three-dimensional Euclidean space that lie at distance r, the radius, from a fixed point. In coordinate geometry a sphere with center (x0, y0, z0) and radius r is the set of all points (x, y, z) such that

(x - x0)2 + (y - y0)2 + (z - z0)2 = r2

A sphere can also be defined as the surface of revolution formed by rotating a circle about its diameter. If the circle is replaced by an ellipse, the shape becomes a spheroid.

The surface area of a sphere is 4πr2 and its volume is 4πr3/3. The sphere has the smallest surface area among all surfaces enclosing a given volume and it encloses the largest volume among all closed surfaces with a given surface area. In nature, bubbles and water drops tend to form spheres because surface tension always tries to minimize surface area.

The intersection of a sphere and any plane is circular; should the plane pass through the center, the intersection is a great circle.

The circumscribed cylinder (see circumscription) for a given sphere has a volume which is 3/2 times the volume of the sphere. If a spherical egg were cut up by an egg-slicer with evenly spaced wires, the bands between the cuts (on the surface of the sphere) would have exactly the same area. Spheres can be generalized to other dimensions. For any natural number n, an n-sphere is the set of points in (n + 1)-dimensional Euclidean space that lie at distance r from a fixed point of that space. A 2-sphere is therefore an ordinary sphere, while a 1-sphere is a circle and a 0-sphere is a pair of points. An n-sphere for which n = 3 or more is often called a hypersphere.


Related entry

   • spherometer


Related categories

   • TOPOLOGY
   • SOLIDS AND SURFACES