# Construct a Dihedral Angle of a Tetrahedron Given Its Plane Angles at a Vertex

Construct a Dihedral Angle of a Tetrahedron Given Its Plane Angles at a Vertex

Given the tetrahedron , at the vertex let be the three plane angles , , . This Demonstration constructs the dihedral angle at the edge .

T=A'B'C'D'

A'

α,β,γ

B'A'C'

B'A'D'

C'A'D'

A'D'

Construct the tetrahedron from the original tetrahedron by modifying the lengths of the sides starting at : cut off by the plane orthogonal to the edge at the vertex . The dihedral angle along is the angle at vertex of the triangle .

ABCD

T

A=A'

T

AD'

D=D'

AD

D

BCD

Given the edge length of and the angles , the other edges are given by: , , , , . It helps to construct the net of .

d

AB

α,β,γ

a=dcos(β)

b=dsin(β)

f=a/cos(γ)

c=fsin(γ)

e=+-2dfcos(α)

2

d

2

f

ABCD