Theorem on the Dihedral Angles of an Isosceles Tetrahedron

tetrahedron

general

isosceles

edge lengths

a

1.1

b

1

c

1

d

e

f

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show spheres

a	b	c	d	e	f	sum
0.13262	0.43369	0.43369	0.43369	0.13262	0.43369	2.

Let the edge lengths of an isosceles tetrahedron

T

be

a

,

b

,

c

,

d=c

,

e=a

and

f=b

. This Demonstration shows that the sum of the cosines of the dihedral angles in

T

equals 2. This is not true for a general tetrahedron, in which opposite sides need not be equal. The derivation makes use of the inscribed and circumscribed spheres of the tetrahedron. In the table,

a

stands for the cosine of the dihedral angle between the two opposite sides ABC and DBC, and so on. For an isosceles tetrahedron, the centers of the inscribed and circumscribed spheres coincide.