TOPICS
Search

Midradius


Midsphere

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

 r_dR=rho^2.
(1)

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

r_d^2=[acsc(pi/p)]^2+R^2
(2)
=a^2+rho^2
(3)
rho^2=[acot(pi/p)]^2+R^2.
(4)

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

r_d=1/2acos(1/2theta)cot(1/2theta)
(5)
rho=1/2acot(1/2theta)
(6)
R=1/2acsc(1/2theta),
(7)

so

 r_d:rho:R=cos(1/2theta):1:sec(1/2theta)
(8)

(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

rho=1/2sqrt(2)sqrt(r_d^2+r_dsqrt(r_d^2+a^2))
(9)
=sqrt(R^2-1/4a^2)
(10)

and these radii obey

 Rr_d=rho^2.
(11)

See also

Archimedean Dual, Archimedean Solid, Circumradius, Inradius, Midsphere, Platonic Solid

Explore with Wolfram|Alpha

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 Web Resource. https://mathworld.wolfram.com/Midradius.html

Subject classifications