Editing Triangle inequality (section)

===Generalization to higher dimensions===
{{unreferenced section|date=October 2021}}

{{see also|Distance geometry#Cayley–Menger determinants}}

The area of a triangular face of a [[tetrahedron]] is less than or equal to the sum of the areas of the other three triangular faces.  More generally, in Euclidean space the hypervolume of an {{math|(''n'' − 1)}}-[[Facet (mathematics)|facet]] of an {{mvar|n}}-[[simplex]] is less than or equal to the sum of the hypervolumes of the other {{mvar|n}} facets.

Much as the triangle inequality generalizes to a polygon inequality, the inequality for a simplex of any dimension generalizes to a [[polytope]] of any dimension: the hypervolume of any facet of a polytope is less than or equal to the sum of the hypervolumes of the remaining facets.

In some cases the tetrahedral inequality is stronger than several applications of the triangle inequality.  For example, the triangle inequality appears to allow the possibility of four points {{mvar|A}}, {{mvar|B}}, {{mvar|C}}, and {{mvar|Z}} in Euclidean space such that distances
:{{math|1=''AB'' = ''BC'' = ''CA'' = 26}}
and
:{{math|1=''AZ'' = ''BZ'' = ''CZ'' = 14}}.
However, points with such distances cannot exist: the area of the {{math|26–26–26}} equilateral triangle {{math|''ABC''}} is <math display=inline>169\sqrt 3</math>, which is larger than three times <math display=inline>39\sqrt 3</math>, the area of a {{math|26–14–14}} isosceles triangle (all by [[Heron's formula]]), and so the arrangement is forbidden by the tetrahedral inequality.