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Zone

Zone

The surface area of a spherical segment. Call the radius of the sphere R, the upper and lower radii b and a, respectively, and the height of the spherical segment h. The zone is a surface of revolution about the z-axis, so the surface area is given by

S=2piintxsqrt(1+x^('2))dz.
(1)

In the xz-plane, the equation of the zone is simply that of a circle,

x=sqrt(R^2-z^2),
(2)

so

x^'=-z(R^2-z^2)^(-1/2)
(3)
x^('2)=(z^2)/(R^2-z^2),
(4)

and

S=2piint_(sqrt(R^2-a^2))^(sqrt(R^2-b^2))sqrt(R^2-z^2)sqrt(1+(z^2)/(R^2-z^2))dz
(5)
=2piRint_(sqrt(R^2-a^2))^(sqrt(R^2-b^2))dz
(6)
=2piR(sqrt(R^2-b^2)-sqrt(R^2-a^2))
(7)
=2piRh.
(8)

This result is somewhat surprising since it depends only on the height of the zone, not its vertical position with respect to the sphere.

See also

Sphere, Spherical Cap, Spherical Segment, Zonohedron

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 130, 1987.Kern, W. F. and Bland, J. R. "Zone." §35 in Solid Mensuration with Proofs, 2nd ed. New York: Wiley, pp. 95-97, 1948.

Cite this as:

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

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