
An elliptic cylinder
In
mathematics, a 'cylinder' is a
quadric surface, with the following equation in
Cartesian coordinates:
:
This equation is for an 'elliptic cylinder', a generalization of the ordinary, 'circular cylinder' (a = b). Even more general is the 'generalized cylinder': the
cross-section can be any curve.
The cylinder is a ''degenerate
quadric'' because at least one of the coordinates (in this case ''z'') does not appear in the equation. By some definitions the cylinder is not considered to be a quadric at all.
In common usage, a ''cylinder'' is taken to mean a finite section of a right circular cylinder with its ends closed to form two circular surfaces, as in the figure (right). If the cylinder has a
radius ''r'' and length (height) ''h'', then its
volume is given by
:
and its
surface area is
:
For a given volume, the cylinder with the smallest surface area has ''h'' = 2''r''. For a given surface area, the cylinder with the largest volume has ''h'' = 2''r'', i.e. the cylinder fits in a cube (height = diameter.)
There are other more unusual types of cylinders. These are the ''imaginary elliptic cylinders'':
:
the ''hyperbolic cylinder'':
:
and the ''parabolic cylinder'':
:
See also
★
Steinmetz solid, the intersection of two or three perpendicular cylinders
★
Prism (geometry)
External links
★
Surface Area MATHguide
★
Volume MATHguide
★
Spinning Cylinder Math Is Fun
★
calculate surface area and volume with your own values
★
Paper model cylinder