geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...

, a spherical segment is the

solid Solid is a state of matter where molecules are closely packed and can not slide past each other. Solids resist compression, expansion, or external forces that would alter its shape, with the degree to which they are resisted dependent upon the ...

defined by cutting a

sphere A sphere (from Ancient Greek, Greek , ) is a surface (mathematics), surface analogous to the circle, a curve. In solid geometry, a sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...

or a

ball A ball is a round object (usually spherical, but sometimes ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used for s ...

with a pair of

parallel planes In geometry, parallel lines are coplanar infinite straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. '' Parallel curves'' are curves that do not touch each oth ...

. It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum. The

surface A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is ...

of the ''spherical segment'' (excluding the bases) is called spherical zone. If the

radius In classical geometry, a radius (: radii or radiuses) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The radius of a regular polygon is th ...

of the sphere is called , the radii of the spherical segment bases are and , and the height of the segment (the distance from one parallel plane to the other) called , then the

volume Volume is a measure of regions in three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch) ...

of the spherical segment is :

V = \frac h \left(3 a^2 + 3 b^2 + h^2\right).

For the special case of the top plane being tangent to the sphere, we have

b = 0

and the solid reduces to a spherical cap. The equation above for volume of the spherical segment can be arranged to :

V = \biggl \pi a^2 \left (\frac \biggr ) \right + \biggl \pi b^2 \left ( \frac \biggr ) \right + \biggl \frac \pi \left( \frac \right)^3 \biggr /math>

Thus, the segment volume equals the sum of three volumes: two right circular cylinders  one of radius  and the second of radius  (both of height h/2) and a sphere of radius h/2 .

The curved

surface area The surface area (symbol ''A'') of a solid object is a measure of the total area that the surface of the object occupies. The mathematical definition of surface area in the presence of curved surfaces is considerably more involved than the d ...

of the spherical zone—which excludes the top and bottom bases—is given by :

A = 2 \pi R h.

Thus the surface area of the segment depends only on the distance between the cutting planes, and not their absolute heights.

References

External links

* *
Summary of spherical formulas
Spherical geometry {{geometry-stub

See also

References

External links