Tetrahedron
Geometry
Level
4
The angle subtended by an edge of a regular tetrahedron at its center is
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1 solution
WLOG a regular tetrahedron can be looked upon as formed by connecting six face diagonals of an unit cube, each length= 2 .
The cube and the tetrahedron share the same center. Thus the distances of the center and all cube vertices= 2 3 .
This is also true for tetrahedron, and is the half solid diagonal.
Thus angle 'A' by an tetrahedron edge at the center is the vertex angle of the isosceles Δ 2 3 , 2 3 , 2 .
By Cos Rule, A = C o s − 1 2 ∗ ( 2 3 ) ∗ ( 2 3 ) ( 2 3 ) 2 + ( 2 3 ) 2 − ( 2 ) 2 = C o s − 1 ( 3 − 1 ) .