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Formeln von verschiedenen Figuren

Trapez

Seiten

\[ a = c + x + y \]\[ b = \frac{h}{\sin(\beta)} \]
\[ d = \sqrt{h^2 + y^2} \]\[ h = \sin({\alpha}) \cdot d \]
\[ c = a - x - y \ \ \ \text{oder} \ \ \ c = \sqrt{h^2 + x^2} \]
\[ x = \sqrt{d^2 - h^2} \]\[ y = \sqrt{b^2 - h^2} \]

Winkel

\[ \alpha = 180° - \delta \]\[ \gamma = 180° - \beta \]
\[ \beta = 180° - \gamma \]\[ \delta = 180° - \alpha \]

Rechteck

Seiten

\[a = d \cdot \sin\left(\frac{f}{2}\right)\]\[b = \sqrt{d^2 - a^2}\]
\[d = \sqrt{a^2 - b^2}\]

Winkel

\[\phi = \arctan\left(\frac{b}{a}\right) \]

Raute

Seiten

\[ a = \sqrt{\left(\frac{e}{2}\right)^2 + \left(\frac{f}{2}\right)^2 } \]
\[ f = 2 \cdot a \cdot \sin\left(\frac{\alpha}{2}\right) \]\[ e = 2 \cdot a \cdot \cos\left(\frac{\alpha}{2}\right) \]

Winkel

\[ \alpha = 2 \cdot \arctan\left(\frac{\frac{f}{2}}{\frac{e}{2}}\right) \]\[ \beta = 180° - \alpha \]

Deltoid

Seiten

\[ e_1 = \sqrt{a^2 - \left(\frac{f}{2}\right)^2} \]\[ e_2 = e - e_1 \]
\[ b = \sqrt{e_2^2 - \left(\frac{f}{2}\right)^2} \]

Winkel

\[ \alpha = \arcsin\left(\frac{\frac{f}{2}}{a}\right) \cdot 2 \]\[ \gamma = \arcsin\left(\frac{\frac{f}{2}}{b}\right) \cdot 2 \]
\[ \beta + \delta = 360° - \alpha - \gamma \]