In geometry, a spherical cap, spherical dome, or spherical segment of one base is a portion of a sphere cut off by a plane. If the plane passes through the center of the sphere, 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
If the radius of the base of the cap is
a
, and the height of the cap is
h
, then the volume of the spherical cap is
V
=
π
h
6
(
3
a
2
+
h
2
)
and the curved surface area of the spherical cap is
A
=
2
π
r
h
or
A
=
2
π
r
2
(
1
−
cos
θ
)
The relationship between
h
and
r
is irrelevant as long as
0
≤
h
≤
2
r
. The red section of the illustration is also a spherical cap.
The parameters
a
,
h
and
r
are not independent:
r
2
=
(
r
−
h
)
2
+
a
2
=
r
2
+
h
2
−
2
r
h
+
a
2
r
=
a
2
+
h
2
2
h
Substituting this into the area formula gives:
A
=
2
π
(
a
2
+
h
2
)
2
h
h
=
π
(
a
2
+
h
2
)
.
Note also that in the upper hemisphere of the diagram,
h
=
r
−
r
2
−
a
2
, and in the lower hemisphere
h
=
r
+
r
2
−
a
2
; hence in either hemisphere
a
=
h
(
2
r
−
h
)
and so an alternative expression for the volume is
V
=
π
h
2
3
(
3
r
−
h
)
.
The volume may also be found by integrating under a surface of rotation, using
x
=
r
cos
(
θ
)
and factorizing.
V
=
∫
x
r
π
(
r
2
−
x
2
)
d
x
=
π
(
2
3
r
3
−
r
2
x
+
1
3
x
3
)
=
π
3
r
3
(
cos
(
θ
)
+
2
)
(
cos
(
θ
)
−
1
)
2
.
Volumes of union and intersection of two intersecting spheres
The volume of the union of two intersecting spheres of radii r1 and r2 is
V
=
V
(
1
)
−
V
(
2
)
,
where
V
(
1
)
=
4
π
3
r
1
3
+
4
π
3
r
2
3
is the sum of the volumes of the two isolated spheres, and
V
(
2
)
=
π
h
1
2
3
(
3
r
1
−
h
1
)
+
π
h
2
2
3
(
3
r
2
−
h
2
)
the sum of the volumes of the two spherical caps forming their intersection. If d < r1 + r2 is the distance between the two sphere centers, elimination of the variables h1 and h2 leads to
V
(
2
)
=
π
12
d
(
r
1
+
r
2
−
d
)
2
(
d
2
+
2
d
(
r
1
+
r
2
)
−
3
(
r
1
−
r
2
)
2
)
.
The surface area bounded by two circles of latitude is the difference of surface areas of their respective spherical caps. For a sphere of radius r, and latitudes φ1 and φ2, the area is
A
=
2
π
r
2
|
sin
ϕ
1
−
sin
ϕ
2
|
For example, assuming the Earth is a sphere of radius 6371 km, the surface area of the arctic (north of the Arctic Circle, at latitude 66.56° as of August 2016) is 2π·6371²|sin 90° − sin 66.56°| = 21.04 million km², or 0.5·|sin 90° − sin 66.56°| = 4.125% of the total surface area of the Earth.
The spheroidal dome is obtained by sectioning off a portion of a spheroid so that the resulting dome is circularly symmetric (having an axis of rotation), and likewise the ellipsoidal dome is derived from the ellipsoid.
Generally, the
n
-dimensional volume of a hyperspherical cap of height
h
and radius
r
in
n
-dimensional Euclidean space is given by
V
=
π
n
−
1
2
r
n
Γ
(
n
+
1
2
)
∫
0
arccos
(
r
−
h
r
)
sin
n
(
t
)
d
t
where
Γ
(the gamma function) is given by
Γ
(
z
)
=
∫
0
∞
t
z
−
1
e
−
t
d
t
.
The formula for
V
can be expressed in terms of the volume of the unit n-ball
C
n
=
π
n
/
2
/
Γ
[
1
+
n
2
]
and the hypergeometric function
2
F
1
or the regularized incomplete beta function
I
x
(
a
,
b
)
as
V
=
C
n
r
n
(
1
2
−
r
−
h
r
Γ
[
1
+
n
2
]
π
Γ
[
n
+
1
2
]
2
F
1
(
1
2
,
1
−
n
2
;
3
2
;
(
r
−
h
r
)
2
)
)
=
1
2
C
n
r
n
I
(
2
r
h
−
h
2
)
/
r
2
(
n
+
1
2
,
1
2
)
,
and the area formula
A
can be expressed in terms of the area of the unit n-ball
A
n
=
2
π
n
/
2
/
Γ
[
n
2
]
as
A
=
1
2
A
n
r
n
−
1
I
(
2
r
h
−
h
2
)
/
r
2
(
n
−
1
2
,
1
2
)
,
where
0
≤
h
≤
r
.
Earlier in (1986, USSR Academ. Press) the following formulas were derived:
A
=
A
n
p
n
−
2
(
q
)
,
V
=
C
n
p
n
(
q
)
, where
q
=
1
−
h
/
r
(
0
≤
q
≤
1
)
,
p
n
(
q
)
=
(
1
−
G
n
(
q
)
/
G
n
(
1
)
)
/
2
,
G
n
(
q
)
=
∫
0
q
(
1
−
t
2
)
(
n
−
1
)
/
2
d
t
.
For odd
n
=
2
k
+
1
:
G
n
(
q
)
=
∑
i
=
0
k
(
−
1
)
i
(
k
i
)
q
2
i
+
1
2
i
+
1
.
It is shown in that, if
n
→
∞
and
q
n
=
const.
, then
p
n
(
q
)
→
1
−
F
(
q
n
)
where
F
(
)
is the integral of the standard normal distribution.
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