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

α

1.

β

0.7

γ

0.5

d

1.2

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Given the tetrahedron

T=A'B'C'D'

, at the vertex

A'

let

α,β,γ

be the three plane angles

B'A'C'

,

B'A'D'

,

C'A'D'

. This Demonstration constructs the dihedral angle at the edge

A'D'

.

Construct the tetrahedron

ABCD

from the original tetrahedron

T

by modifying the lengths of the sides starting at

A=A'

: cut

T

off by the plane orthogonal to the edge

AD'

at the vertex

D=D'

. The dihedral angle along

AD

is the angle at vertex

D

of the triangle

BCD

.

Given the edge length

d

of

AB

and the angles

α,β,γ

, the other edges are given by:

a=dcos(β)

,

b=dsin(β)

,

f=a/cos(γ)

,

c=fsin(γ)

,

e=

2

d

+

2

f

-2dfcos(α)

. It helps to construct the net of

ABCD

.