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) |
This parameter generalizes the identity
 |
(2) |
where 
is the volume, 
is the inradius, and 
is the surface area, which
is valid only for symmetrical solids, to
 |
(3) |
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) |
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) |
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) |
is root of a high-order polynomial, and
![h_5=1/(1202)[1/(10)(121461425+53168861sqrt(5)-4sqrt(30(28343974350325+12675597513679sqrt(5)))]^(1/2).](https://txtdot.deep-swarm.xyz/proxy/img?url=https%3A%2F%2Fmathworld.wolfram.com%2Fimages%2Fequations%2FHarmonicParameter%2FNumberedEquation7.svg) |
(7) |
See also
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References
Fjelstad, P. and Ginchev, I. "Volume, Surface Area, and the Harmonic Mean." Math. Mag. 76, 126-129, 2003.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Harmonic Parameter." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/HarmonicParameter.html