In
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 cap or spherical dome is a portion of 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 of 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 ...
cut off by a
plane. It is also a
spherical segment
In geometry, a spherical segment is the solid defined by cutting a sphere or a ball with a pair of parallel planes.
It can be thought of as a spherical cap with the top truncated, and so it corresponds to a spherical frustum.
The surface o ...
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.
Discussion
Any arc of a great circle is a geodesic of the sphere, so that great circles in spher ...
), so that the height of the cap is equal to 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, the spherical cap is called a ''
hemisphere
Hemisphere may refer to:
In geometry
* Hemisphere (geometry), a half of a sphere
As half of Earth or any spherical astronomical object
* A hemisphere of Earth
** Northern Hemisphere
** Southern Hemisphere
** Eastern Hemisphere
** Western Hemi ...
''.
Volume and surface area
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 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 forming the base of the cap.
These variables are inter-related through the formulas
,
,
,
and
.
If
denotes the
latitude
In geography, latitude is a geographic coordinate system, geographic coordinate that specifies the north-south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from −90° at t ...
in
geographic coordinates
A geographic coordinate system (GCS) is a spherical or geodetic coordinate system for measuring and communicating positions directly on Earth as latitude and longitude. It is the simplest, oldest, and most widely used type of the various ...
, then
, and
.
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
Area is the measure of a region's size on a surface. The area of a plane region or ''plane area'' refers to the area of a shape or planar lamina, while '' surface area'' refers to the area of an open surface or the boundary of a three-di ...
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