structural:reformatting, linguistic:paraphrasing
Higher dimensions

A tetrahedron has four triangular facets. The absolute value of the polar sine (psin) of the normal vectors to the three facets that share a vertex of the tetrahedron, divided by the area of the fourth facet will not depend upon the choice of the vertex:

| psin (b, c, d) | A r e a a = | psin (a, c, d) | A r e a b = | psin (a, b, d) | A r e a c = | psin (a, b, c) | A r e a d = (3 V o l u m e t e t r a h e d r o n) 2 2 A r e a a A r e a b A r e a c A r e a d. {\displaystyle {\begin{aligned}&{\frac {\left|\operatorname {psin} (\mathbf {b},\mathbf {c},\mathbf {d})\right|}{\mathrm {Area} _{a}}}={\frac {\left|\operatorname {psin} (\mathbf {a},\mathbf {c},\mathbf {d})\right|}{\mathrm {Area} _{b}}}={\frac {\left|\operatorname {psin} (\mathbf {a},\mathbf {b},\mathbf {d})\right|}{\mathrm {Area} _{c}}}={\frac {\left|\operatorname {psin} (\mathbf {a},\mathbf {b},\mathbf {c})\right|}{\mathrm {Area} _{d}}}\\={}&{\frac {(3~\mathrm {Volume} _{\mathrm {tetrahedron} })^{2}}{2~\mathrm {Area} _{a}\mathrm {Area} _{b}\mathrm {Area} _{c}\mathrm {Area} _{d}}}\,.\end{aligned}}}

More generally, for an n -dimensional simplex (i.e., triangle (n = 2), tetrahedron (n = 3), pentatope (n = 4), etc.) in n -dimensional Euclidean space, the absolute value of the polar sine of the normal vectors of the facets that meet at a vertex, divided by the hyperarea of the facet opposite the vertex is independent of the choice of the vertex. Writing V for the hypervolume of the n -dimensional simplex and P for the product of the hyperareas of its (n − 1) -dimensional facets, the common ratio is | psin (b, …, z) | A r e a a = ⋯ = | psin (a, …, y) | A r e a z = (n V) n − 1 (n − 1) ! P. {\displaystyle {\frac {\left|\operatorname {psin} (\mathbf {b},\ldots,\mathbf {z})\right|}{\mathrm {Area} _{a}}}=\cdots ={\frac {\left|\operatorname {psin} (\mathbf {a},\ldots,\mathbf {y})\right|}{\mathrm {Area} _{z}}}={\frac {(nV)^{n-1}}{(n-1)!P}}.}