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Spherical Segment


SphericalSegment

A spherical segment is the solid defined by cutting a sphere 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 a zone. However, Harris and Stocker (1998) use the term "spherical segment" as a synonym for spherical cap and "zone" for what is here called a spherical segment.

Call the radius of the sphere R and the height of the segment (the distance from the plane to the top of sphere) h. Let the radii of the lower and upper bases be denoted a and b, respectively. Call the distance from the center to the start of the segment d, and the height from the bottom to the top of the segment h. Call the radius parallel to the segment r, and the height above the center y. Then r^2=R^2-y^2,

V=int_d^(d+h)pir^2dy
(1)
=piint_d^(d+h)(R^2-y^2)dy
(2)
=pi[R^2y-1/3y^3]_d^(d+h)
(3)
=pi{R^2h-1/3[(d+h)^3-d^3]}
(4)
=pi[R^2h-1/3(d^3+3d^2h+3h^2d+h^3-d^3)]
(5)
=pih(R^2-d^2-hd-1/3h^2).
(6)

Relationships among the various quantities include

a=sqrt(R^2-d^2)
(7)
b=sqrt(R^2-(d+h)^2)
(8)
=sqrt(R^2-d^2-2dh-h^2)
(9)
d=(a^2-b^2-h^2)/(2h)
(10)
R=sqrt(([(a-b)^2+h^2][(a+b)^2+h^2])/(4h^2)).
(11)

Plugging in gives

V=pih[1/2(a^2+b^2+h^2)-1/3h^2]
(12)
=pih(1/2a^2+1/2b^2+1/6h^2)
(13)
=1/6pih(3a^2+3b^2+h^2).
(14)

The surface area of the zone (which excludes the top and bottom bases) is given by

 S=2piRh.
(15)

See also

Archimedes' Hat-Box Theorem, Archimedes' Problem, Frustum, Hemisphere, Sphere, Spherical Cap, Spherical Sector, Spherical Wedge, Surface of Revolution, Zone

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References

Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 130, 1987.Harris, J. W. and Stocker, H. "Spherical Zone (Spherical Layer)." §4.8.5 in Handbook of Mathematics and Computational Science. New York: Springer-Verlag, pp. 107-108, 1998.Kern, W. F. and Bland, J. R. "Spherical Segment." §36 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 97-102, 1948.Smith, D. E. "Spherical Segment." §541 in Essentials of Plane and Solid Geometry. Boston, MA: Ginn and Co., p. 542, 1923.

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Spherical Segment

Cite this as:

Weisstein, Eric W. "Spherical Segment." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/SphericalSegment.html

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