Teilaufgabe a

Satz des Pythagoras  anwenden

${\displaystyle h^2+{\left(\frac{\sqrt{2}}{2}a\right)}^2=a^2\Rightarrow h=\frac{\sqrt{2}}{2}a}\backslash displaystyle\; h^2+\backslash left(\backslash frac\{\backslash sqrt2\}2a\backslash right)^2=a^2\backslash ;\backslash Rightarrow\backslash ;h=\backslash frac\{\backslash sqrt2\}2a$

Teilaufgabe c

Die Höhe innerhalb der Dreiecks-Seitenfläche bestimmen. Satz des Pythagoras anwenden.

${\displaystyle {\left(h_{△}\right)}^2+{\left(\frac{1}{2}a\right)}^2=a^2\Rightarrow h_{△}=\frac{\sqrt{3}}{2}a}\backslash displaystyle\; \backslash left(h\_\backslash bigtriangleup\backslash right)^2+\backslash left(\backslash frac12a\backslash right)^2=a^2\backslash ;\backslash Rightarrow\backslash ;h\_\backslash bigtriangleup=\backslash frac\{\backslash sqrt3\}2a$

Flächeninhalt des Dreiecks bestimmen:

$A_{△}=\frac{1}{2}\cdot a\cdot h_{△}=\frac{\sqrt{3}}{4}{a}^{2}A\_\backslash bigtriangleup=\backslash frac12\backslash cdot\; a\backslash cdot\; h\_\backslash bigtriangleup=\backslash frac\{\backslash sqrt3\}4a^2$

Oberflächeninhalt bestimmen:

$O=4\cdot A_{△}+a^2=a^2+4\cdot \frac{\sqrt{3}}{4}{a}^2=(1+\sqrt{3})\cdot a^{2}O=4\backslash cdot\; A\_\backslash bigtriangleup+a^2=a^2+4\backslash cdot\backslash frac\{\backslash sqrt3\}4a^2=\backslash left(1+\backslash sqrt3\backslash right)\backslash cdot\; a^2$