spherical segment

{{Short description|Region between parallel planes intersecting a sphere}}

{{inline |date=May 2024}}

File:LaoHaiKugelschicht1.png

File:01sphere2planes.pdf

File:02sph.seg.3D.pdf

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 of the spherical segment (excluding the bases) is called spherical zone.

File:Geometric parameters for spherical segment.pdf

If the radius of the sphere is called {{mvar|R}}, the radii of the spherical segment bases are {{mvar|a}} and {{mvar|b}}, and the height of the segment (the distance from one parallel plane to the other) called {{mvar|h}}, then the volume of the spherical segment is

: $V\; =\; \backslash frac\{\backslash pi\}\{6\}\; h\; \backslash left(3\; a^2\; +\; 3\; b^2\; +\; h^2\backslash right).$

For the special case of the top plane being tangent to the sphere, we have $b\; =\; 0$ and the solid reduces to a spherical cap.{{cite book |last1=Kern |first1=Willis |last2=Bland |first2=James |title=Solid Mensuration with Proofs |date=1938 |publisher=John Wiley & Sons, Inc |location=New York |pages=97-103 |edition=Second |url=https://archive.org/details/in.ernet.dli.2015.205959 |access-date=16 May 2024}}

The equation above for volume of the spherical segment can be arranged to

: $V\; =\; \backslash biggl\; [\; \backslash pi\; a^2\; \backslash left\; (\backslash frac\{h\}\{2\}\; \backslash biggr\; )\; \backslash right\; ]\; +\; \backslash biggl\; [\; \backslash pi\; b^2\; \backslash left\; (\; \backslash frac\{h\}\{2\}\; \backslash biggr\; )\; \backslash right\; ]\; +\; \backslash biggl\; [\; \backslash frac\{4\}\{3\}\; \backslash pi\; \backslash left(\; \backslash frac\{h\}\{2\}\; \backslash right)^3\; \backslash biggr\; ]$

Thus, the segment volume equals the sum of three volumes: two right circular cylinders one of radius {{mvar|a}} and the second of radius {{mvar|b}} (both of height $h/2$) and a sphere of radius $h/2$.

The curved surface area of the spherical zone—which excludes the top and bottom bases—is given by

: $A\; =\; 2\; \backslash pi\; R\; h.$

Thus the surface area of the segment depends only on the distance between the cutting planes, and not their absolute heights.