A spherical layer , also known as a spherical disk, is a part of a sphere that is cut out by two parallel planes . The curved surface part is called the spherical zone .
Formulas
The following formulas apply to the calculation of volume , surface area and surface of a spherical layer . It denotes the radius of the sphere , the radii of the boundary circles and the height of the spherical layer.
r
{\ displaystyle r}
a
1
,
a
2
{\ displaystyle a_ {1}, a_ {2}}
H
{\ displaystyle h}
These three quantities are not independent of one another. The spherical layer is determined by any three of these four sizes. The fourth can be calculated from three of the four quantities.
Sizes of a spherical segment with the radius r of the sphere, the radii a 1 , a 2 of the boundary circles and the heights h , h 1 , h 2
volume
V
=
π
3
⋅
(
H
1
-
H
2
)
⋅
(
3
⋅
(
H
1
+
H
2
)
⋅
r
-
(
H
1
2
+
H
1
⋅
H
2
+
H
2
2
)
)
{\ displaystyle V = {\ frac {\ pi} {3}} \ cdot (h_ {1} -h_ {2}) \ cdot (3 \ cdot (h_ {1} + h_ {2}) \ cdot r- (h_ {1} ^ {2} + h_ {1} \ cdot h_ {2} + h_ {2} ^ {2}))}
V
=
π
6th
⋅
H
⋅
(
3
⋅
a
1
2
+
3
⋅
a
2
2
+
H
2
)
{\ displaystyle V = {\ frac {\ pi} {6}} \ cdot h \ cdot (3 \ cdot a_ {1} ^ {2} +3 \ cdot a_ {2} ^ {2} + h ^ {2 })}
Area of the lateral surface
M.
=
2
⋅
π
⋅
r
⋅
H
{\ displaystyle M = 2 \ cdot \ pi \ cdot r \ cdot h}
M.
=
2
⋅
π
⋅
H
⋅
(
a
1
2
+
(
a
1
2
-
a
2
2
-
H
2
2
⋅
H
)
2
)
1
2
{\ displaystyle M = 2 \ cdot \ pi \ cdot h \ cdot \ left (a_ {1} ^ {2} + \ left ({\ frac {a_ {1} ^ {2} -a_ {2} ^ {2 } -h ^ {2}} {2 \ cdot h}} \ right) ^ {2} \ right) ^ {\ frac {1} {2}}}
Surface area
O
=
π
⋅
(
2
⋅
r
⋅
H
+
a
1
2
+
a
2
2
)
{\ displaystyle O = \ pi \ cdot (2 \ cdot r \ cdot h + a_ {1} ^ {2} + a_ {2} ^ {2})}
O
=
π
⋅
(
2
⋅
H
⋅
(
a
1
2
+
(
a
1
2
-
a
2
2
-
H
2
2
⋅
H
)
2
)
1
2
+
a
1
2
+
a
2
2
)
{\ displaystyle O = \ pi \ cdot (2 \ cdot h \ cdot \ left (a_ {1} ^ {2} + \ left ({\ frac {a_ {1} ^ {2} -a_ {2} ^ { 2} -h ^ {2}} {2 \ cdot h}} \ right) ^ {2} \ right) ^ {\ frac {1} {2}} + a_ {1} ^ {2} + a_ {2 } ^ {2})}
Derivation
The spherical layer can be imagined as the spherical segment with the lower circle as the base circle, from which the spherical segment with the upper circle as the base circle is taken away. Let it be the height of and the height of . The volumes of the two spherical segments are (see spherical segment ). So is
S.
1
{\ displaystyle {\ mathcal {S}} _ {1}}
S.
2
{\ displaystyle {\ mathcal {S}} _ {2}}
H
1
{\ displaystyle h_ {1}}
S.
1
{\ displaystyle {\ mathcal {S}} _ {1}}
H
2
{\ displaystyle h_ {2}}
S.
2
{\ displaystyle {\ mathcal {S}} _ {2}}
V
1
=
π
3
⋅
H
1
2
⋅
(
3
⋅
r
-
H
1
)
,
V
2
=
π
3
⋅
H
2
2
⋅
(
3
⋅
r
-
H
2
)
{\ displaystyle V_ {1} = {\ frac {\ pi} {3}} \ cdot h_ {1} ^ {2} \ cdot (3 \ cdot r-h_ {1}), \ V_ {2} = { \ frac {\ pi} {3}} \ cdot h_ {2} ^ {2} \ cdot (3 \ cdot r-h_ {2})}
V
=
V
1
-
V
2
=
π
3
⋅
(
3
⋅
(
H
1
2
-
H
2
2
)
⋅
r
-
(
H
1
3
-
H
2
3
)
)
=
π
3
⋅
(
H
1
-
H
2
)
⋅
(
3
⋅
(
H
1
+
H
2
)
⋅
r
-
(
H
1
2
+
H
1
⋅
H
2
+
H
2
2
)
)
{\ displaystyle {\ begin {aligned} V & = V_ {1} -V_ {2} = {\ frac {\ pi} {3}} \ cdot (3 \ cdot (h_ {1} ^ {2} -h_ { 2} ^ {2}) \ cdot r- (h_ {1} ^ {3} -h_ {2} ^ {3})) \\ & = {\ frac {\ pi} {3}} \ cdot (h_ {1} -h_ {2}) \ cdot (3 \ cdot (h_ {1} + h_ {2}) \ cdot r- (h_ {1} ^ {2} + h_ {1} \ cdot h_ {2} + h_ {2} ^ {2})) \ end {aligned}}}
With the relationships (see spherical segment ) results
2
⋅
r
⋅
H
1
=
a
1
2
+
H
1
2
,
2
⋅
r
⋅
H
2
=
a
2
2
+
H
2
2
{\ displaystyle 2 \ cdot r \ cdot h_ {1} = a_ {1} ^ {2} + h_ {1} ^ {2}, \ 2 \ cdot r \ cdot h_ {2} = a_ {2} ^ { 2} + h_ {2} ^ {2}}
V
=
π
3
⋅
(
H
1
-
H
2
)
⋅
(
3
2
⋅
(
a
1
2
+
H
1
2
+
a
2
2
+
H
2
2
)
-
H
1
2
-
H
1
⋅
H
2
-
H
2
2
)
=
π
6th
⋅
(
H
1
-
H
2
)
⋅
(
3
⋅
(
a
1
2
+
a
2
2
)
+
(
H
1
-
H
2
)
2
)
{\ displaystyle {\ begin {aligned} V & = {\ frac {\ pi} {3}} \ cdot (h_ {1} -h_ {2}) \ cdot \ left ({\ frac {3} {2}} \ cdot (a_ {1} ^ {2} + h_ {1} ^ {2} + a_ {2} ^ {2} + h_ {2} ^ {2}) - h_ {1} ^ {2} -h_ {1} \ cdot h_ {2} -h_ {2} ^ {2} \ right) \\ & = {\ frac {\ pi} {6}} \ cdot (h_ {1} -h_ {2}) \ cdot (3 \ cdot (a_ {1} ^ {2} + a_ {2} ^ {2}) + (h_ {1} -h_ {2}) ^ {2}) \ end {aligned}}}
There follows the formula above:
H
=
H
1
-
H
2
{\ displaystyle h = h_ {1} -h_ {2}}
V
=
π
6th
⋅
H
⋅
(
3
⋅
a
1
2
+
3
⋅
a
2
2
+
H
2
)
{\ displaystyle V = {\ frac {\ pi} {6}} \ cdot h \ cdot (3 \ cdot a_ {1} ^ {2} +3 \ cdot a_ {2} ^ {2} + h ^ {2 })}
The same results
for the outer surface
M.
=
M.
1
-
M.
2
=
2
⋅
π
⋅
r
⋅
H
1
-
2
⋅
π
⋅
r
⋅
H
2
=
2
⋅
π
⋅
r
⋅
(
H
1
-
H
2
)
=
2
⋅
π
⋅
r
⋅
H
{\ displaystyle M = M_ {1} -M_ {2} = 2 \ cdot \ pi \ cdot r \ cdot h_ {1} -2 \ cdot \ pi \ cdot r \ cdot h_ {2} = 2 \ cdot \ pi \ cdot r \ cdot (h_ {1} -h_ {2}) = 2 \ cdot \ pi \ cdot r \ cdot h}
Relationship of parameters
To prove the relationship between let the distance of the lower plane to the center of the sphere . Then applies
r
,
a
1
,
a
2
,
H
{\ displaystyle r, a_ {1}, a_ {2}, h}
d
{\ displaystyle d}
M.
{\ displaystyle M}
r
2
=
d
2
+
a
1
2
,
r
2
=
(
d
+
H
)
2
+
a
2
2
{\ displaystyle r ^ {2} = d ^ {2} + a_ {1} ^ {2}, \ r ^ {2} = (d + h) ^ {2} + a_ {2} ^ {2}}
If one sets the two equations equally and solves for , one obtains and with the first equation follows
d
{\ displaystyle d}
d
=
a
1
2
-
a
2
2
-
H
2
2
⋅
H
{\ displaystyle d = {\ frac {a_ {1} ^ {2} -a_ {2} ^ {2} -h ^ {2}} {2 \ cdot h}}}
r
2
=
a
1
2
+
(
a
1
2
-
a
2
2
-
H
2
2
⋅
H
)
2
{\ displaystyle r ^ {2} = a_ {1} ^ {2} + \ left ({\ frac {a_ {1} ^ {2} -a_ {2} ^ {2} -h ^ {2}} { 2 \ cdot h}} \ right) ^ {2}}
See also
literature
I. Bronstein et al. a .: Paperback of mathematics. Harri Deutsch, Frankfurt 2001, ISBN 3-8171-2005-2 .
Small Encyclopedia Mathematics , Harri Deutsch-Verlag, 1977, p. 215.
L. Kusch u. a .: Mathematics, part 4 integral calculus. Cornelsen, Berlin 2000, ISBN 3-464-41304-7 .
Web links
<img src="https://de.wikipedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" title="" width="1" height="1" style="border: none; position: absolute;">