Die Kohte entspricht einem gleichseitigem Achteck.
Inkreisradius
r
i
=
a
1
2
(
1
+
2
)
{\displaystyle r_{i}=a\ {\frac {1}{2}}(1+{\sqrt {2}})}
a
=
2
r
i
(
2
−
1
)
{\displaystyle a=2r_{i}\ ({\sqrt {2}}-1)}
Umkreisradius
r
u
=
a
1
2
4
+
2
2
{\displaystyle r_{u}=a\ {\frac {1}{2}}{\sqrt {4+2{\sqrt {2}}}}}
r
u
=
a
1
+
1
2
{\displaystyle r_{u}=a\ {\sqrt {1+{\frac {1}{\sqrt {2}}}}}}
a
=
r
u
2
−
2
{\displaystyle a=r_{u}\ {\sqrt {2-{\sqrt {2}}}}}
Große Diagonale
d
1
=
a
4
+
2
2
=
2
r
u
{\displaystyle d_{1}=a\ {\sqrt {4+2{\sqrt {2}}}}\ =\ 2r_{u}}
Mittlere Diagonale
d
2
=
a
(
1
+
2
)
=
2
r
i
{\displaystyle d_{2}=a\ (1+{\sqrt {2}})\ =\ 2r_{i}}
Kleine Diagonale
d
3
=
a
2
+
2
=
r
u
2
{\displaystyle d_{3}=a\ {\sqrt {2+{\sqrt {2}}}}\ =\ r_{u}\,{\sqrt {2}}}
Zentriwinkel
α
=
360
∘
8
=
45
∘
{\displaystyle \alpha ={\frac {360^{\circ }}{8}}=45^{\circ }}
Innenwinkel
δ
=
180
∘
−
α
=
135
∘
{\displaystyle \delta =180^{\circ }-\alpha =135^{\circ }}
cos
δ
=
−
1
2
{\displaystyle \cos \delta ={\frac {-1}{\sqrt {2}}}}
Flächeninhalt
A
=
a
2
(
2
+
2
2
)
{\displaystyle A=a^{2}\ (2+2{\sqrt {2}})}
A
=
r
u
2
2
2
{\displaystyle A=r_{u}^{2}\ 2{\sqrt {2}}}
Zentriwinkel
α
=
360
∘
12
=
30
∘
{\displaystyle \alpha ={\frac {360^{\circ }}{12}}=30^{\circ }}
Innenwinkel
δ
=
180
∘
−
α
=
150
∘
{\displaystyle \delta =180^{\circ }-\alpha =150^{\circ }}
Flächeninhalt
A
=
3
cot
(
π
12
)
a
2
=
3
(
2
+
3
)
a
2
≈
11,196
15
a
2
.
{\displaystyle {\begin{aligned}A&=3\cot \left({\frac {\pi }{12}}\right)a^{2}=3\left(2+{\sqrt {3}}\right)a^{2}\\&\approx 11{,}19615\,a^{2}.\end{aligned}}}
Die Fläche kann auch mit
R
{\displaystyle R}
Umkreises [1]
A
=
6
sin
(
π
6
)
R
2
=
3
R
2
.
{\displaystyle A=6\sin \left({\frac {\pi }{6}}\right)R^{2}=3R^{2}.}
Mit r als Radius des Inkreises , ergibt sich der Flächeninhalt des regelmäßigen Zwölfecks zu
A
=
12
tan
(
π
12
)
r
2
=
12
(
2
−
3
)
r
2
≈
3,215
39
r
2
.
{\displaystyle {\begin{aligned}A&=12\tan \left({\frac {\pi }{12}}\right)r^{2}=12\left(2-{\sqrt {3}}\right)r^{2}\\&\approx 3{,}21539\,r^{2}.\end{aligned}}}
