Beispiel Nr: 11
$ \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=8m \qquad h=9m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(8m)^2+(8m)^2} =11,3m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{8m}{2}\right)^2+(9m)^2} =9,85m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{11,3m}{2}\right)^2+(9m)^2} =10,6m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 8m \cdot 9,85m =158m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (8m)^2=64m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 64m^{2}+158m^{2}=222m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (8m)^2 \cdot 9m =192m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{9m}{\frac{1}{2}11,3m} \\ \eta=57,8 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{9m}{\frac{1}{2}8m} \\ \epsilon=66^{\circ} \\ \\\\\\ \tiny \begin{array}{|l|} \hline a=\\ \hline 8 m \\ \hline 80 dm \\ \hline 800 cm \\ \hline 8\cdot 10^{3} mm \\ \hline 8\cdot 10^{6} \mu m \\ \hline 0,008 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h=\\ \hline 9 m \\ \hline 90 dm \\ \hline 900 cm \\ \hline 9\cdot 10^{3} mm \\ \hline 9\cdot 10^{6} \mu m \\ \hline 0,009 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline V=\\ \hline 192 m^3 \\ \hline 1,92\cdot 10^{5} dm^3 \\ \hline 1,92\cdot 10^{8} cm^3 \\ \hline 1,92\cdot 10^{11} mm^3 \\ \hline 1,92\cdot 10^{5} l \\ \hline 1,92\cdot 10^{3} hl \\ \hline 1,92\cdot 10^{8} ml \\ \hline \end{array} \tiny \begin{array}{|l|} \hline d=\\ \hline 11,3 m \\ \hline 113 dm \\ \hline 1,13\cdot 10^{3} cm \\ \hline 1,13\cdot 10^{4} mm \\ \hline 1,13\cdot 10^{7} \mu m \\ \hline 0,0113 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline h1=\\ \hline 9,85 m \\ \hline 98,5 dm \\ \hline 985 cm \\ \hline 9,85\cdot 10^{3} mm \\ \hline 9,85\cdot 10^{6} \mu m \\ \hline 0,00985 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 9,85 m \\ \hline 98,5 dm \\ \hline 985 cm \\ \hline 9,85\cdot 10^{3} mm \\ \hline 9,85\cdot 10^{6} \mu m \\ \hline 0,00985 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 10,6 m \\ \hline 106 dm \\ \hline 1,06\cdot 10^{3} cm \\ \hline 1,06\cdot 10^{4} mm \\ \hline 1,06\cdot 10^{7} \mu m \\ \hline 0,0106 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline M=\\ \hline 158 m^2 \\ \hline 1,58\cdot 10^{4} dm^2 \\ \hline 1,58\cdot 10^{6} cm^2 \\ \hline 1,58\cdot 10^{8} mm^2 \\ \hline 1,58 a \\ \hline 0,0158 ha \\ \hline 0,000158 km^2 \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline G=\\ \hline 64 m^2 \\ \hline 6,4\cdot 10^{3} dm^2 \\ \hline 6,4\cdot 10^{5} cm^2 \\ \hline 6,4\cdot 10^{7} mm^2 \\ \hline 0,64 a \\ \hline 0,0064 ha \\ \hline 6,4\cdot 10^{-5} km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 222 m^3 \\ \hline 2,22\cdot 10^{5} dm^3 \\ \hline 2,22\cdot 10^{8} cm^3 \\ \hline 2,22\cdot 10^{11} mm^3 \\ \hline 2,22\cdot 10^{5} l \\ \hline 2,22\cdot 10^{3} hl \\ \hline 2,22\cdot 10^{8} ml \\ \hline \end{array}$