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Rob Marshall
Artikel: Spherical cap GudangMovies21 Rebahinxxi
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball 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), 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
r
{\displaystyle r}
of the sphere
The radius
a
{\displaystyle a}
of the base of the cap
The height
h
{\displaystyle h}
of the cap
The polar angle
θ
{\displaystyle \theta }
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
a
=
r
sin
θ
{\displaystyle a=r\sin \theta }
,
h
=
r
(
1
−
cos
θ
)
{\displaystyle h=r(1-\cos \theta )}
,
2
h
r
=
a
2
+
h
2
{\displaystyle 2hr=a^{2}+h^{2}}
,
and
2
h
a
=
(
a
2
+
h
2
)
sin
θ
{\displaystyle 2ha=(a^{2}+h^{2})\sin \theta }
.
If
ϕ
{\displaystyle \phi }
denotes the latitude in geographic coordinates, then
θ
+
ϕ
=
π
/
2
=
90
∘
{\displaystyle \theta +\phi =\pi /2=90^{\circ }\,}
, and
cos
θ
=
sin
ϕ
{\displaystyle \cos \theta =\sin \phi }
.
= 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
V
s
e
c
{\displaystyle V_{sec}}
of the spherical sector, by an intuitive argument, as
A
=
3
r
V
s
e
c
=
3
r
2
π
r
2
h
3
=
2
π
r
h
.
{\displaystyle A={\frac {3}{r}}V_{sec}={\frac {3}{r}}{\frac {2\pi r^{2}h}{3}}=2\pi rh\,.}
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
V
=
1
3
b
h
′
{\displaystyle V={\frac {1}{3}}bh'}
, where
b
{\displaystyle b}
is the infinitesimal area of each pyramidal base (located on the surface of the sphere) and
h
′
{\displaystyle h'}
is the height of each pyramid from its base to its apex (at the center of the sphere). Since each
h
′
{\displaystyle h'}
, in the limit, is constant and equivalent to the radius
r
{\displaystyle r}
of the sphere, the sum of the infinitesimal pyramidal bases would equal the area of the spherical sector, and:
V
s
e
c
=
∑
V
=
∑
1
3
b
h
′
=
∑
1
3
b
r
=
r
3
∑
b
=
r
3
A
{\displaystyle V_{sec}=\sum {V}=\sum {\frac {1}{3}}bh'=\sum {\frac {1}{3}}br={\frac {r}{3}}\sum b={\frac {r}{3}}A}
= Deriving the volume and surface area using calculus
=The volume and area formulas may be derived by examining the rotation of the function
f
(
x
)
=
r
2
−
(
x
−
r
)
2
=
2
r
x
−
x
2
{\displaystyle f(x)={\sqrt {r^{2}-(x-r)^{2}}}={\sqrt {2rx-x^{2}}}}
for
x
∈
[
0
,
h
]
{\displaystyle x\in [0,h]}
, using the formulas the surface of the rotation for the area and the solid of the revolution for the volume.
The area is
A
=
2
π
∫
0
h
f
(
x
)
1
+
f
′
(
x
)
2
d
x
{\displaystyle A=2\pi \int _{0}^{h}f(x){\sqrt {1+f'(x)^{2}}}\,dx}
The derivative of
f
{\displaystyle f}
is
f
′
(
x
)
=
r
−
x
2
r
x
−
x
2
{\displaystyle f'(x)={\frac {r-x}{\sqrt {2rx-x^{2}}}}}
and hence
1
+
f
′
(
x
)
2
=
r
2
2
r
x
−
x
2
{\displaystyle 1+f'(x)^{2}={\frac {r^{2}}{2rx-x^{2}}}}
The formula for the area is therefore
A
=
2
π
∫
0
h
2
r
x
−
x
2
r
2
2
r
x
−
x
2
d
x
=
2
π
∫
0
h
r
d
x
=
2
π
r
[
x
]
0
h
=
2
π
r
h
{\displaystyle A=2\pi \int _{0}^{h}{\sqrt {2rx-x^{2}}}{\sqrt {\frac {r^{2}}{2rx-x^{2}}}}\,dx=2\pi \int _{0}^{h}r\,dx=2\pi r\left[x\right]_{0}^{h}=2\pi rh}
The volume is
V
=
π
∫
0
h
f
(
x
)
2
d
x
=
π
∫
0
h
(
2
r
x
−
x
2
)
d
x
=
π
[
r
x
2
−
1
3
x
3
]
0
h
=
π
h
2
3
(
3
r
−
h
)
{\displaystyle V=\pi \int _{0}^{h}f(x)^{2}\,dx=\pi \int _{0}^{h}(2rx-x^{2})\,dx=\pi \left[rx^{2}-{\frac {1}{3}}x^{3}\right]_{0}^{h}={\frac {\pi h^{2}}{3}}(3r-h)}
Applications
= Volumes of union and intersection of two intersecting spheres
=The volume of the union of two intersecting spheres
of radii
r
1
{\displaystyle r_{1}}
and
r
2
{\displaystyle r_{2}}
is
V
=
V
(
1
)
−
V
(
2
)
,
{\displaystyle V=V^{(1)}-V^{(2)}\,,}
where
V
(
1
)
=
4
π
3
r
1
3
+
4
π
3
r
2
3
{\displaystyle V^{(1)}={\frac {4\pi }{3}}r_{1}^{3}+{\frac {4\pi }{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
)
{\displaystyle V^{(2)}={\frac {\pi h_{1}^{2}}{3}}(3r_{1}-h_{1})+{\frac {\pi h_{2}^{2}}{3}}(3r_{2}-h_{2})}
the sum of the volumes of the two spherical caps forming their intersection. If
d
≤
r
1
+
r
2
{\displaystyle d\leq r_{1}+r_{2}}
is the
distance between the two sphere centers, elimination of the variables
h
1
{\displaystyle h_{1}}
and
h
2
{\displaystyle h_{2}}
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
)
.
{\displaystyle V^{(2)}={\frac {\pi }{12d}}(r_{1}+r_{2}-d)^{2}\left(d^{2}+2d(r_{1}+r_{2})-3(r_{1}-r_{2})^{2}\right)\,.}
= Volume of a spherical cap with a curved base
=The volume of a spherical cap with a curved base can be calculated by considering two spheres with radii
r
1
{\displaystyle r_{1}}
and
r
2
{\displaystyle r_{2}}
, separated by some distance
d
{\displaystyle d}
, and for which their surfaces intersect at
x
=
h
{\displaystyle x=h}
. That is, the curvature of the base comes from sphere 2. The volume is thus the difference between sphere 2's cap (with height
(
r
2
−
r
1
)
−
(
d
−
h
)
{\displaystyle (r_{2}-r_{1})-(d-h)}
) and sphere 1's cap (with height
h
{\displaystyle h}
),
V
=
π
h
2
3
(
3
r
1
−
h
)
−
π
[
(
r
2
−
r
1
)
−
(
d
−
h
)
]
2
3
[
3
r
2
−
(
(
r
2
−
r
1
)
−
(
d
−
h
)
)
]
,
V
=
π
h
2
3
(
3
r
1
−
h
)
−
π
3
(
d
−
h
)
3
(
r
2
−
r
1
d
−
h
−
1
)
2
[
2
r
2
+
r
1
d
−
h
+
1
]
.
{\displaystyle {\begin{aligned}V&={\frac {\pi h^{2}}{3}}(3r_{1}-h)-{\frac {\pi [(r_{2}-r_{1})-(d-h)]^{2}}{3}}[3r_{2}-((r_{2}-r_{1})-(d-h))]\,,\\V&={\frac {\pi h^{2}}{3}}(3r_{1}-h)-{\frac {\pi }{3}}(d-h)^{3}\left({\frac {r_{2}-r_{1}}{d-h}}-1\right)^{2}\left[{\frac {2r_{2}+r_{1}}{d-h}}+1\right]\,.\end{aligned}}}
This formula is valid only for configurations that satisfy
0
<
d
<
r
2
{\displaystyle 0
and
d
−
(
r
2
−
r
1
)
<
h
≤
r
1
{\displaystyle d-(r_{2}-r_{1})
. If sphere 2 is very large such that
r
2
≫
r
1
{\displaystyle r_{2}\gg r_{1}}
, hence
d
≫
h
{\displaystyle d\gg h}
and
r
2
≈
d
{\displaystyle r_{2}\approx d}
, which is the case for a spherical cap with a base that has a negligible curvature, the above equation is equal to the volume of a spherical cap with a flat base, as expected.
= Areas of intersecting spheres
=Consider two intersecting spheres of radii
r
1
{\displaystyle r_{1}}
and
r
2
{\displaystyle r_{2}}
, with their centers separated by distance
d
{\displaystyle d}
. They intersect if
|
r
1
−
r
2
|
≤
d
≤
r
1
+
r
2
{\displaystyle |r_{1}-r_{2}|\leq d\leq r_{1}+r_{2}}
From the law of cosines, the polar angle of the spherical cap on the sphere of radius
r
1
{\displaystyle r_{1}}
is
cos
θ
=
r
1
2
−
r
2
2
+
d
2
2
r
1
d
{\displaystyle \cos \theta ={\frac {r_{1}^{2}-r_{2}^{2}+d^{2}}{2r_{1}d}}}
Using this, the surface area of the spherical cap on the sphere of radius
r
1
{\displaystyle r_{1}}
is
A
1
=
2
π
r
1
2
(
1
+
r
2
2
−
r
1
2
−
d
2
2
r
1
d
)
{\displaystyle A_{1}=2\pi r_{1}^{2}\left(1+{\frac {r_{2}^{2}-r_{1}^{2}-d^{2}}{2r_{1}d}}\right)}
= Surface area bounded by parallel disks
=The curved surface area of the spherical segment bounded by two parallel disks is the difference of surface areas of their respective spherical caps. For a sphere of radius
r
{\displaystyle r}
, and caps with heights
h
1
{\displaystyle h_{1}}
and
h
2
{\displaystyle h_{2}}
, the area is
A
=
2
π
r
|
h
1
−
h
2
|
,
{\displaystyle A=2\pi r|h_{1}-h_{2}|\,,}
or, using geographic coordinates with latitudes
ϕ
1
{\displaystyle \phi _{1}}
and
ϕ
2
{\displaystyle \phi _{2}}
,
A
=
2
π
r
2
|
sin
ϕ
1
−
sin
ϕ
2
|
,
{\displaystyle A=2\pi r^{2}|\sin \phi _{1}-\sin \phi _{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 (8.12 million sq mi), or 0.5 ⋅ |sin 90° − sin 66.56°| = 4.125% of the total surface area of the Earth.
This formula can also be used to demonstrate that half the surface area of the Earth lies between latitudes 30° South and 30° North in a spherical zone which encompasses all of the Tropics.
Generalizations
= Sections of other solids
=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.
= Hyperspherical cap
=Generally, the
n
{\displaystyle n}
-dimensional volume of a hyperspherical cap of height
h
{\displaystyle h}
and radius
r
{\displaystyle r}
in
n
{\displaystyle 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
(
θ
)
d
θ
{\displaystyle V={\frac {\pi ^{\frac {n-1}{2}}\,r^{n}}{\,\Gamma \left({\frac {n+1}{2}}\right)}}\int _{0}^{\arccos \left({\frac {r-h}{r}}\right)}\sin ^{n}(\theta )\,\mathrm {d} \theta }
where
Γ
{\displaystyle \Gamma }
(the gamma function) is given by
Γ
(
z
)
=
∫
0
∞
t
z
−
1
e
−
t
d
t
{\displaystyle \Gamma (z)=\int _{0}^{\infty }t^{z-1}\mathrm {e} ^{-t}\,\mathrm {d} t}
.
The formula for
V
{\displaystyle V}
can be expressed in terms of the volume of the unit n-ball
C
n
=
π
n
/
2
/
Γ
[
1
+
n
2
]
{\textstyle C_{n}=\pi ^{n/2}/\Gamma [1+{\frac {n}{2}}]}
and the hypergeometric function
2
F
1
{\displaystyle {}_{2}F_{1}}
or the regularized incomplete beta function
I
x
(
a
,
b
)
{\displaystyle 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
)
,
{\displaystyle V=C_{n}\,r^{n}\left({\frac {1}{2}}\,-\,{\frac {r-h}{r}}\,{\frac {\Gamma [1+{\frac {n}{2}}]}{{\sqrt {\pi }}\,\Gamma [{\frac {n+1}{2}}]}}{\,\,}_{2}F_{1}\left({\tfrac {1}{2}},{\tfrac {1-n}{2}};{\tfrac {3}{2}};\left({\tfrac {r-h}{r}}\right)^{2}\right)\right)={\frac {1}{2}}C_{n}\,r^{n}I_{(2rh-h^{2})/r^{2}}\left({\frac {n+1}{2}},{\frac {1}{2}}\right),}
and the area formula
A
{\displaystyle A}
can be expressed in terms of the area of the unit n-ball
A
n
=
2
π
n
/
2
/
Γ
[
n
2
]
{\textstyle A_{n}={2\pi ^{n/2}/\Gamma [{\frac {n}{2}}]}}
as
A
=
1
2
A
n
r
n
−
1
I
(
2
r
h
−
h
2
)
/
r
2
(
n
−
1
2
,
1
2
)
,
{\displaystyle A={\frac {1}{2}}A_{n}\,r^{n-1}I_{(2rh-h^{2})/r^{2}}\left({\frac {n-1}{2}},{\frac {1}{2}}\right),}
where
0
≤
h
≤
r
{\displaystyle 0\leq h\leq r}
.
A. Chudnov derived the following formulas:
A
=
A
n
r
n
−
1
p
n
−
2
(
q
)
,
V
=
C
n
r
n
p
n
(
q
)
,
{\displaystyle A=A_{n}r^{n-1}p_{n-2}(q),\,V=C_{n}r^{n}p_{n}(q),}
where
q
=
1
−
h
/
r
(
0
≤
q
≤
1
)
,
p
n
(
q
)
=
(
1
−
G
n
(
q
)
/
G
n
(
1
)
)
/
2
,
{\displaystyle q=1-h/r(0\leq q\leq 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
.
{\displaystyle G_{n}(q)=\int _{0}^{q}(1-t^{2})^{(n-1)/2}dt.}
For odd
n
=
2
k
+
1
{\displaystyle n=2k+1}
:
G
n
(
q
)
=
∑
i
=
0
k
(
−
1
)
i
(
k
i
)
q
2
i
+
1
2
i
+
1
.
{\displaystyle G_{n}(q)=\sum _{i=0}^{k}(-1)^{i}{\binom {k}{i}}{\frac {q^{2i+1}}{2i+1}}.}
Asymptotics
If
n
→
∞
{\displaystyle n\to \infty }
and
q
n
=
const.
{\displaystyle q{\sqrt {n}}={\text{const.}}}
, then
p
n
(
q
)
→
1
−
F
(
q
n
)
{\displaystyle p_{n}(q)\to 1-F({q{\sqrt {n}}})}
where
F
(
)
{\displaystyle F()}
is the integral of the standard normal distribution.
A more quantitative bound is
A
/
(
A
n
r
n
−
1
)
=
n
Θ
(
1
)
⋅
[
(
2
−
h
/
r
)
h
/
r
]
n
/
2
{\displaystyle A/(A_{n}r^{n-1})=n^{\Theta (1)}\cdot [(2-h/r)h/r]^{n/2}}
.
For large caps (that is when
(
1
−
h
/
r
)
4
⋅
n
=
O
(
1
)
{\displaystyle (1-h/r)^{4}\cdot n=O(1)}
as
n
→
∞
{\displaystyle n\to \infty }
), the bound simplifies to
n
Θ
(
1
)
⋅
e
−
(
1
−
h
/
r
)
2
n
/
2
{\displaystyle n^{\Theta (1)}\cdot e^{-(1-h/r)^{2}n/2}}
.
See also
Circular segment — the analogous 2D object
Solid angle — contains formula for n-sphere caps
Spherical segment
Spherical sector
Spherical wedge
References
Further reading
Richmond, Timothy J. (1984). "Solvent accessible surface area and excluded volume in proteins: Analytical equation for overlapping spheres and implications for the hydrophobic effect". Journal of Molecular Biology. 178 (1): 63–89. doi:10.1016/0022-2836(84)90231-6. PMID 6548264.
Lustig, Rolf (1986). "Geometry of four hard fused spheres in an arbitrary spatial configuration". Molecular Physics. 59 (2): 195–207. Bibcode:1986MolPh..59..195L. doi:10.1080/00268978600102011.
Gibson, K. D.; Scheraga, Harold A. (1987). "Volume of the intersection of three spheres of unequal size: a simplified formula". The Journal of Physical Chemistry. 91 (15): 4121–4122. doi:10.1021/j100299a035.
Gibson, K. D.; Scheraga, Harold A. (1987). "Exact calculation of the volume and surface area of fused hard-sphere molecules with unequal atomic radii". Molecular Physics. 62 (5): 1247–1265. Bibcode:1987MolPh..62.1247G. doi:10.1080/00268978700102951.
Petitjean, Michel (1994). "On the analytical calculation of van der Waals surfaces and volumes: some numerical aspects". Journal of Computational Chemistry. 15 (5): 507–523. doi:10.1002/jcc.540150504.
Grant, J. A.; Pickup, B. T. (1995). "A Gaussian description of molecular shape". The Journal of Physical Chemistry. 99 (11): 3503–3510. doi:10.1021/j100011a016.
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External links
Weisstein, Eric W. "Spherical cap". MathWorld. Derivation and some additional formulas.
Online calculator for spherical cap volume and area.
Summary of spherical formulas.
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Artikel Terkait "spherical cap"
Spherical cap - Wikipedia
In geometry, a spherical cap or spherical dome is a portion of a sphere or of a ball cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane.
Spherical Cap - Wolfram MathWorld
5 days ago · A spherical cap is the region of a sphere which lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is a called a hemisphere, and if the cap is cut by a second plane, the spherical frustum is called a spherical segment.
How to Calculate the Volume of a Spherical Cap - Maths at Home
The volume of a spherical cap in terms of ‘r’ and ‘h’ is V = (πh2/3) (3r-h), where r is the radius of the sphere and h is the height of the spherical cap.
Spherical Cap volume and surface area, calculator, formula
A spherical cap or spherical dome is a portion of a sphere cut off by a plane. It is also a spherical segment of one base, i.e., bounded by a single plane. To perform the calculation, enter the spherical radius and the height or radius of the cap. Then press the 'Calculate' button. V s = 1 3 ⋅π ⋅h2 ⋅(3 ⋅r− h) V s = 1 3 · π · h 2 · (3 · r − h)
Spherical cap and spherical segment - Online calculators
This calculator computes volume and surface area of spherical cap and spherical segment. A spherical cap is the region of a sphere that lies above (or below) a given plane. If the plane passes through the center of the sphere, the cap is called a hemisphere.
Sphere, Spherical Sector, Spherical cap, Spherical segment
A spherical sector is a portion of a sphere defined by a conical boundary with apex at the center of the sphere. A spherical cap is a portion of a sphere cut off by a plane. A spherical segment is a portion of the sphere included between two parallel planes.
Spherical Cap Volume Formula with Solved Examples
07 Agu 2024 · It is the section of a sphere that extends above the sphere’s plane and formed when a plane cuts off a part of a sphere. The base area, height, and sphere radius are all the values that are required to calculate the volume of a spherical cap. Below is the formula for Spherical Cap Volume. π is a constant with a value of 22/7.
Spherical Cap - Michigan State University
26 Mei 1999 · A spherical cap is the region of a Sphere which lies above (or below) a given Plane. If the Plane passes through the Center of the Sphere, the cap is a Hemisphere.
Spherical Cap Calculator
Are you tired of manually calculating the volume and surface area of spherical caps? Look no further than our Spherical Cap Calculator! Whether you're a student learning about geometry or a professional in need of a quick calculation, our tool provides accurate results in just a few clicks.
Spherical Cap Calculator
Calculate the volume, arc length, base and surface area of a spherical cap. This calculator works for any measurement unit.