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Hemisphere -- from Wolfram MathWorld

  • ️Weisstein, Eric W.

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Hemisphere

Half of a sphere cut by a plane passing through its center. A hemisphere of radius r can be given by the usual spherical coordinates

x=rcosthetasinphi

(1)

y=rsinthetasinphi

(2)

z=rcosphi,

(3)

where theta in [0,2pi) and phi in [0,pi/2]. All cross sections passing through the z-axis are semicircles.

The volume of the hemisphere is

V=int_0^rint_0^(pi/2)int_0^(2pi)rho^2cosphidthetadphidrho

(4)

=piint_0^r(r^2-z^2)dz

(5)

=2/3pir^3.

(6)

The weighted mean of z over the hemisphere is

<z>=piint_0^rz(r^2-z^2)dz=1/4pir^4.

(7)

The geometric centroid is then given by

z^_=(<z>)/V=3/8r

(8)

(Beyer 1987).

See also

Capsule, Semicircle, Sphere

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References

Beyer, W. H. (Ed.). CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, p. 133, 1987.

Cite this as:

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

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