In
geometry, a spherical cap or spherical dome is a portion of a
sphere or of a
ball
A ball is a round object (usually spherical, but can sometimes be 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 f ...
cut off by a
plane. It is also a
spherical segment of one base, i.e., bounded by a single plane. If the plane passes through the
center of the sphere (forming a
great circle
In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point.
Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geomet ...
), so that the height of the cap is equal to the
radius of the sphere, the spherical cap is called a ''
hemisphere''.
Volume and surface area
The
volume of the spherical cap and the area of the curved surface may be calculated using combinations of
* The radius
of the sphere
* The radius
of the base of the cap
* The height
of the cap
* The
polar angle between the rays from the center of the sphere to the apex of the cap (the pole) and the edge of the
disk
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
forming the base of the cap
If
denotes the
latitude in
geographic coordinates, then
, and
.
The relationship between
and
is relevant as long as
. For example, the red section of the illustration is also a spherical cap for which
.
The formulas using
and
can be rewritten to use the radius
of the base of the cap instead of
, using the
Pythagorean theorem
In mathematics, the Pythagorean theorem or Pythagoras' theorem is a fundamental relation in Euclidean geometry between the three sides of a right triangle. It states that the area of the square whose side is the hypotenuse (the side opposite t ...
:
:
so that
:
Substituting this into the formulas gives:
:
:
Deriving the surface area intuitively from the spherical sector volume
Note that aside from the calculus based argument below, the area of the spherical cap may be derived from the volume
of the
spherical sector, by an intuitive argument, as
:
The intuitive argument is based upon summing the total sector volume from that of infinitesimal
triangular pyramids. Utilizing the
pyramid (or cone) volume formula of
, where
is the infinitesimal
area of each pyramidal base (located on the surface of the sphere) and
is the height of each pyramid from its base to its apex (at the center of the sphere). Since each
, in the limit, is constant and equivalent to the radius
of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:
:
Deriving the volume and surface area using calculus
The volume and area formulas may be derived by examining the rotation of the function
:
for