Beispiel Nr: 13
$\text{Gegeben:}\\ \text{Länge der Seite } \qquad a \qquad [m] \\ \text{Körperhöhe } \qquad h \qquad [m] \\ \\ \text{Gesucht:} \\ \text{Diagonale } \qquad d \qquad [m] \\ \text{Seitenkante } \qquad s \qquad [m] \\ \text{Grundfläche} \qquad G \qquad [m^{2}] \\ \text{Mantelfläche} \qquad M \qquad [m^{2}] \\ \text{Volumen} \qquad V \qquad [m^{3}] \\ \\ \text{Quadratische Pyramide}\\ \textbf{Gegeben:} \\ a=13m \qquad h=16m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(13m)^2+(13m)^2} =18,4m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{13m}{2}\right)^2+(16m)^2} =17,3m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{18,4m}{2}\right)^2+(16m)^2} =18,5m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 13m \cdot 17,3m =449m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (13m)^2=169m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 169m^{2}+449m^{2}=618m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (13m)^2 \cdot 16m =901\frac{1}{3}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{16m}{\frac{1}{2}18,4m} \\ \eta=60,1 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{16m}{\frac{1}{2}13m} \\ \epsilon=67,9^{\circ} \\ \\\\\\ \tiny \begin{array}{|l|} \hline a=\\ \hline 13 m \\ \hline 130 dm \\ \hline 1,3\cdot 10^{3} cm \\ \hline 1,3\cdot 10^{4} mm \\ \hline 1,3\cdot 10^{7} \mu m \\ \hline 0,013 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h=\\ \hline 16 m \\ \hline 160 dm \\ \hline 1,6\cdot 10^{3} cm \\ \hline 1,6\cdot 10^{4} mm \\ \hline 1,6\cdot 10^{7} \mu m \\ \hline 0,016 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline V=\\ \hline 901 m^3 \\ \hline 9,01\cdot 10^{5} dm^3 \\ \hline 9,01\cdot 10^{8} cm^3 \\ \hline 9,01\cdot 10^{11} mm^3 \\ \hline 9,01\cdot 10^{5} l \\ \hline 9,01\cdot 10^{3} hl \\ \hline 9,01\cdot 10^{8} ml \\ \hline \end{array} \tiny \begin{array}{|l|} \hline d=\\ \hline 18,4 m \\ \hline 184 dm \\ \hline 1,84\cdot 10^{3} cm \\ \hline 1,84\cdot 10^{4} mm \\ \hline 1,84\cdot 10^{7} \mu m \\ \hline 0,0184 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline h1=\\ \hline 17,3 m \\ \hline 173 dm \\ \hline 1,73\cdot 10^{3} cm \\ \hline 1,73\cdot 10^{4} mm \\ \hline 1,73\cdot 10^{7} \mu m \\ \hline 0,0173 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 17,3 m \\ \hline 173 dm \\ \hline 1,73\cdot 10^{3} cm \\ \hline 1,73\cdot 10^{4} mm \\ \hline 1,73\cdot 10^{7} \mu m \\ \hline 0,0173 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 18,5 m \\ \hline 185 dm \\ \hline 1,85\cdot 10^{3} cm \\ \hline 1,85\cdot 10^{4} mm \\ \hline 1,85\cdot 10^{7} \mu m \\ \hline 0,0185 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline M=\\ \hline 449 m^2 \\ \hline 4,49\cdot 10^{4} dm^2 \\ \hline 4,49\cdot 10^{6} cm^2 \\ \hline 4,49\cdot 10^{8} mm^2 \\ \hline 4,49 a \\ \hline 0,0449 ha \\ \hline 0,000449 km^2 \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline G=\\ \hline 169 m^2 \\ \hline 1,69\cdot 10^{4} dm^2 \\ \hline 1,69\cdot 10^{6} cm^2 \\ \hline 1,69\cdot 10^{8} mm^2 \\ \hline 1,69 a \\ \hline 0,0169 ha \\ \hline 0,000169 km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 618 m^3 \\ \hline 6,18\cdot 10^{5} dm^3 \\ \hline 6,18\cdot 10^{8} cm^3 \\ \hline 6,18\cdot 10^{11} mm^3 \\ \hline 6,18\cdot 10^{5} l \\ \hline 6,18\cdot 10^{3} hl \\ \hline 6,18\cdot 10^{8} ml \\ \hline \end{array}$