Sines of the Dihedral Angles of a Tetrahedron
Sines of the Dihedral Angles of a Tetrahedron
Let be a tetrahedron with dihedral angles , , , , , . A theorem relating the sines of the dihedral angles states that .
ABCD
θ
AD
θ
BD
θ
CD
θ
AB
θ
BC
θ
AC
AD×BCsinsin=BD×ACsinsin=CD×ABsinsin
θ
AD
θ
BC
θ
BD
θ
AC
θ
CD
θ
AB
Here , and are pairs of opposite edges of the tetrahedron.
(AD,BC)
(BD,AC)
(CD,AB)