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)
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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
(2)
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(3)
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(4)
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Furthermore, let be the angle subtended by the polyhedron edge of an Archimedean solid. Then
(5)
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(6)
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(7)
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so
(8)
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(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
(9)
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(10)
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and these radii obey
(11)
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