The radius 
of the midsphere of a polyhedron,
also called the interradius. Let 
be a point on the original polyhedron and 
the corresponding point 
on the dual. Then because 
and 
are inverse points, the radii 
, 
, and 
satisfy
 |
(1) |
The above figure shows a plane section of a midsphere.
Let 
be the inradius
the dual polyhedron, 
circumradius of the original polyhedron, and 
the side length of the original polyhedron. For a regular
polyhedron with Schläfli symbol 
, the dual
polyhedron is 
.
Then
Furthermore, let 
be the angle subtended by the polyhedron
edge of an Archimedean solid. Then
so
 |
(8) |
(Cundy and Rollett 1989).
For a Platonic or Archimedean solid, the midradius 
of the solid and dual can be expressed in terms of the circumradius 
of the solid and inradius 
of the dual gives
and these radii obey
 |
(11) |
See also
Archimedean Dual, Archimedean Solid, Circumradius, Inradius, Midsphere, Platonic Solid
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References
Cundy, H. and Rollett, A. Mathematical Models, 3rd ed. Stradbroke, England: Tarquin Pub., pp. 126-127, 1989.
Referenced on Wolfram|Alpha
Cite this as:
Weisstein, Eric W. "Midradius." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Midradius.html