Formelsammlung

Achteck

Inkreisradius

$r_{i}=a\ {\frac {1}{2}}(1+{\sqrt {2}})$

$a=2r_{i}\ ({\sqrt {2}}-1)$

Umkreisradius

$r_{u}=a\ {\frac {1}{2}}{\sqrt {4+2{\sqrt {2}}}}$

$r_{u}=a\ {\sqrt {1+{\frac {1}{\sqrt {2}}}}}$

$a=r_{u}\ {\sqrt {2-{\sqrt {2}}}}$

Große Diagonale

$d_{1}=a\ {\sqrt {4+2{\sqrt {2}}}}\ =\ 2r_{u}$

Mittlere Diagonale

$d_{2}=a\ (1+{\sqrt {2}})\ =\ 2r_{i}$

Kleine Diagonale

$d_{3}=a\ {\sqrt {2+{\sqrt {2}}}}\ =\ r_{u}\,{\sqrt {2}}$

Zentriwinkel

$\alpha ={\frac {360^{\circ }}{8}}=45^{\circ }$

Innenwinkel

|$\delta =180^{\circ }-\alpha =135^{\circ }$

$\cos \delta ={\frac {-1}{\sqrt {2}}}$

Flächeninhalt

$A=a^{2}\ (2+2{\sqrt {2}})$

$A=r_{u}^{2}\ 2{\sqrt {2}}$