Angle between adjacent sides of a regular dodecahedron
Your friend is into Dungeons and Dragons and wants to make fancy dice for the game. He has an access to a faceting machine he wants to use for the purpose, but to program it correctly he needs to know the angle between the adjacent sides of a regular dodecahedron.
Give the angle, in degrees, to three decimal places.
The answer is 116.56505.
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3 solutions
The dihedral angle
θ
of a regular dodecahedron is form by three faces of a regular pentagon.
If the angles of the faces are A,B,C we have,
C
o
s
θ
=
S
i
n
A
∗
S
i
n
B
C
o
s
A
−
C
o
s
B
∗
c
o
s
C
.
B
u
t
a
l
l
t
h
r
e
e
∠
s
o
f
a
r
e
g
u
l
a
r
p
e
n
t
a
g
o
n
a
r
e
5
5
−
2
∗
1
8
0
=
1
0
8
o
.
∴
θ
=
C
o
s
−
1
S
i
n
1
0
8
∗
S
i
n
1
0
8
C
o
s
1
0
8
−
C
o
s
1
0
8
∗
c
o
s
1
0
8
=
1
1
6
.
5
6
5
0
6
o
.
The dihedral angle of a platonic solid composed of regular p − gons, with q meeting at each vertex is given as
sin ( 2 θ ) = sin ( π / p ) cos ( π / q )
In a duodecahedron, three pentagons meet at each vertex. Hence,
θ = 2 sin − 1 ( sin ( π / 5 ) cos ( π / 3 ) ) = 2 sin − 1 ( 0 . 5 8 7 7 0 . 5 ) = 2 . 0 3 4 r a d = 1 1 6 . 5 6 5 ∘
Within the regular pentagon ABCDE the angle BAC is 36 degrees. Form a right triangle ABF within the same plane. Use this to calculate distances x and y (assuming the side to be 1).
x = t a n ( 3 6 ) , y = c o s ( 3 6 ) 1
Similarly form a right triangle ABG in the adjacent side. The triangle AFG in a plane below is equilateral. This is because it is similar to the triangle ACH, with which it lays in the same plane. The triangle BFG, the one which contains the dihedral angle GBF, has sides lengths x, x, and y. From this we get the angle as
2 A r c s i n ( 2 x y )