Exsphere (polyhedra)

The exsphere touches the face of the regular polyedron at the center of the incircle of that face. If the exsphere radius is denoted r_ex, the radius of this incircle r_in and the dihedral angle between the face and the extension of the adjacent face δ, the center of the exsphere is located from the viewpoint at the middle of one edge of the face by bisecting the dihedral angle. Therefore

${\displaystyle \tan {\frac {\delta }{2}}={\frac {r_{\mathrm {ex} }}{r_{\mathrm {in} }}}.}$

δ is the 180-degree complement of the internal face-to-face angle.

Applied to the geometry of the Tetrahedron of edge length a, we have an incircle radius r_in = a/(2√3) (derived by dividing twice the face area (a²√3)/4 through the perimeter 3a), a dihedral angle δ = π - arccos(1/3), and in consequence r_ex = a/√6.

The radius of the exspheres of the 6 faces of the Cube is the same as the radius of the inscribed sphere, since δ and its complement are the same, 90 degrees.

The dihedral angle applicable to the Icosahedron is derived by considering the coordinates of two triangles with a common edge, for example one face with vertices at

${\displaystyle (0,-1,g),(g,0,1),(0,1,g),}$

the other at

${\displaystyle (1,-g,0),(g,0,1),(0,-1,g),}$

where g is the golden ratio. Subtracting vertex coordinates defines edge vectors,

${\displaystyle (g,1,1-g),(-g,1,g-1)}$

of the first face and

${\displaystyle (g-1,g,1),(-g,-1,g-1)}$

of the other. Cross products of the edges of the first face and second face yield (not normalized) face normal vectors

${\displaystyle (2g-2,0,2g)\sim (g-1,0,g)}$

of the first and

${\displaystyle (g^{2}-g+1,-g-(g-1)^{2},1-g+g^{2})=(2,-2,2)\sim (1,-1,1)}$

of the second face, using g²=1+g. The dot product between these two face normals yields the cosine of the dihedral angle,

${\displaystyle \cos \delta ={\frac {(g-1)\cdot 1+g\cdot 1}{{\sqrt {(g-1)^{2}+g^{2}}}{\sqrt {3}}}}={\frac {2g-1}{3}}={\frac {\surd 5}{3}}\approx 0.74535599.}$ OEIS: A208899

${\displaystyle \therefore \delta \approx 0.72973\,\mathrm {rad} \approx 41.81^{\circ }}$

${\displaystyle \therefore \tan {\frac {\delta }{2}}={\frac {\sin \delta }{1+\cos \delta }}={\frac {2}{3+\surd 5}}\approx 0.3819660}$ OEIS: A132338

For an icosahedron of edge length a, the incircle radius of the triangular faces is r_in = a/(2√3), and finally the radius of the 20 exspheres

${\displaystyle r_{\mathrm {ex} }={\frac {a}{(3+{\sqrt {5}}){\sqrt {3}}}}\approx 0.1102641a.}$

• Insphere

• Gerber, Leon (1977). "Associated and skew-orthologic simplexes". Trans. Am. Math. Soc. 231 (1): 47–63. doi:10.1090/S0002-9947-1977-0445393-6. JSTOR 1997867. MR 0445393.
• Hajja, Mowaffaq (2005). "The Gergonne and Nagel centers of an n-dimensional simplex". J. Geom. 83 (1–2): 46–56. doi:10.1007/s00022-005-0011-3. S2CID 123076195.