The harmonic parameter of a polyhedron is the weighted mean of the distances from a fixed interior point to the faces, where the weights are the areas of the faces, i.e.,
(1)
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This parameter generalizes the identity
(2)
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where is the volume, is the inradius, and is the surface area, which is valid only for symmetrical solids, to
(3)
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The harmonic parameter is independent of the choice of interior point (Fjelstad and Ginchev 2003). In addition, it can be defined not only for polyhedron, but any -dimensional solids that have -dimensional content and -dimensional content .
Expressing the area and perimeter of a lamina in terms of gives the identity
(4)
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The following table summarizes the harmonic parameter for a few common laminas. Here, is the inradius of a given lamina, and and are the side lengths of a rectangle.
Expressing and for a solid in terms of then gives the identity
(5)
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The following table summarizes the harmonic parameter for a few common solids, where some of the more complicated values are given by the polynomial roots
(6)
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is root of a high-order polynomial, and
(7)
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