WOLFRAM|DEMONSTRATIONS PROJECT

Sines of the Dihedral Angles of a Tetrahedron

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edge lengths
|AD|
1.18
|BD|
1
|CD|
1
|AB|
1
|BC|
1
|AC|
1
show labels
edge
AD
BD
CD
AB
BC
AC
length
1.1800
1
1
1
1
1
sine of angle
0.9725
0.9066
0.9066
0.9066
0.9974
0.9066
|AD|×|BC|
sin
θ
AD
sin
θ
BC
|BD|×|AC|
sin
θ
BD
sin
θ
AC
|CD|×|AB|
sin
θ
CD
sin
θ
AB
1.2165
1.2165
1.2165
Let
ABCD
be a tetrahedron with dihedral angles
θ
AD
,
θ
BD
,
θ
CD
,
θ
AB
,
θ
BC
,
θ
AC
. A theorem relating the sines of the dihedral angles states that
AD×BCsin
θ
AD
sin
θ
BC
=BD×ACsin
θ
BD
sin
θ
AC
=CD×ABsin
θ
CD
sin
θ
AB
.
Here
(AD,BC)
,
(BD,AC)
and
(CD,AB)
are pairs of opposite edges of the tetrahedron.