MathDB
Tetrahedron with an Inscribed Sphere

Source:

January 11, 2009
geometry3D geometrytetrahedronsphereprobabilityinradiuscircumcircle

Problem Statement

A tetrahedron with four equilateral triangular faces has a sphere inscribed within it and a sphere circumscribed about it. For each of the four faces, there is a sphere tangent externally to the face at its center and to the circumscribed sphere. A point P P is selected at random inside the circumscribed sphere. The probability that P P lies inside one of the five small spheres is closest to <spanclass=latexbold>(A)</span> 0<spanclass=latexbold>(B)</span> 0.1<spanclass=latexbold>(C)</span> 0.2<spanclass=latexbold>(D)</span> 0.3<spanclass=latexbold>(E)</span> 0.4 <span class='latex-bold'>(A)</span>\ 0\qquad <span class='latex-bold'>(B)</span>\ 0.1\qquad <span class='latex-bold'>(C)</span>\ 0.2\qquad <span class='latex-bold'>(D)</span>\ 0.3\qquad <span class='latex-bold'>(E)</span>\ 0.4