# Geometry of the Tetrahedron

Steven Dutch, Natural and Applied Sciences, University of Wisconsin - Green Bay
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Since the tetrahedron occurs so often in crystallography, it's useful to summarize its geometry.

In the diagram above left,

• Edges are defined as equal to unity: AB=BC=CA=AO=BO=CO=1
• ABC is an equilateral triangle, so AE=CD = sqrt(3)/2, etc.
• The medians of a triangle all intersect at a point that divides the medians in ratio 1:2. Thus FE = AE/3 and AF = 2AE/3, etc.
• Since AE = sqrt(3)/2, AE = sqrt(3)/3 and FE = sqrt(3)/6, etc.
• Finally, since AO=1 and AF = sqrt(3)/3, OF = sqrt(2/3) = sqrt(6)/3 = 0.8165.

The altitude of a tetrahedron = sqrt(2/3) = sqrt(6)/3 = 0.8165

The diagram above right illustrates how to find FX.

• OF is perpendicular to AF and AG is perpendicular to OG
• Since they are both right triangles and have angle O in common, triangles OFB and OXG are similar.
• Therefore XG/FB=OG/OF
• By symmetry, XF = XG, so XF/FB = OG/OF
• XF = FB*OG/OF
• FB = sqrt(3)/6, OG = sqrt(3)/3 and OF = sqrt(6)/3
• Therefore XF = sqrt(6)/12 = 0.2041
• Since OF = sqrt(6)/3 and XF = sqrt(6)/12, XF=OF/4

The centroid of a tetrahedron is one-fourth of the way from base to opposing vertex.