Midradius

The radius 👁 rho
of the midsphere of a polyhedron, also called the interradius. Let 👁 P
be a point on the original polyhedron and 👁 P^'
the corresponding point 👁 P
on the dual. Then because 👁 P
and 👁 P^'
are inverse points, the radii 👁 r_d=OP^'
, 👁 R=OP
, and 👁 rho=OQ
satisfy

The above figure shows a plane section of a midsphere.

Let 👁 r_d
be the inradius the dual polyhedron, 👁 R
circumradius of the original polyhedron, and 👁 a
the side length of the original polyhedron. For a regular polyhedron with Schläfli symbol 👁 {q,p}
, the dual polyhedron is 👁 {p,q}
. Then

Furthermore, let 👁 theta
be the angle subtended by the polyhedron edge of an Archimedean solid. Then

so

(Cundy and Rollett 1989).

For a Platonic or Archimedean solid, the midradius 👁 rho=rho_d
of the solid and dual can be expressed in terms of the circumradius 👁 R
of the solid and inradius 👁 r_d
of the dual gives

and these radii obey

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

Midradius

Cite this as:

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

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