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).](/images/equations/HarmonicParameter/NumberedEquation7.svg) |
(7)
|
