Beispiel Nr: 10
$ \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=14m \qquad h=15m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(14m)^2+(14m)^2} =19,8m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{14m}{2}\right)^2+(15m)^2} =16,6m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{19,8m}{2}\right)^2+(15m)^2} =18m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 14m \cdot 16,6m =463m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (14m)^2=196m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 196m^{2}+463m^{2}=659m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (14m)^2 \cdot 15m =980m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{15m}{\frac{1}{2}19,8m} \\ \eta=56,6 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{15m}{\frac{1}{2}14m} \\ \epsilon=65^{\circ} \\ \\\\\\ \tiny \begin{array}{|l|} \hline a=\\ \hline 14 m \\ \hline 140 dm \\ \hline 1,4\cdot 10^{3} cm \\ \hline 1,4\cdot 10^{4} mm \\ \hline 1,4\cdot 10^{7} \mu m \\ \hline 0,014 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h=\\ \hline 15 m \\ \hline 150 dm \\ \hline 1,5\cdot 10^{3} cm \\ \hline 1,5\cdot 10^{4} mm \\ \hline 1,5\cdot 10^{7} \mu m \\ \hline 0,015 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline V=\\ \hline 980 m^3 \\ \hline 9,8\cdot 10^{5} dm^3 \\ \hline 9,8\cdot 10^{8} cm^3 \\ \hline 9,8\cdot 10^{11} mm^3 \\ \hline 9,8\cdot 10^{5} l \\ \hline 9,8\cdot 10^{3} hl \\ \hline 9,8\cdot 10^{8} ml \\ \hline \end{array} \tiny \begin{array}{|l|} \hline d=\\ \hline 19,8 m \\ \hline 198 dm \\ \hline 1,98\cdot 10^{3} cm \\ \hline 1,98\cdot 10^{4} mm \\ \hline 1,98\cdot 10^{7} \mu m \\ \hline 0,0198 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline h1=\\ \hline 16,6 m \\ \hline 166 dm \\ \hline 1,66\cdot 10^{3} cm \\ \hline 1,66\cdot 10^{4} mm \\ \hline 1,66\cdot 10^{7} \mu m \\ \hline 0,0166 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 16,6 m \\ \hline 166 dm \\ \hline 1,66\cdot 10^{3} cm \\ \hline 1,66\cdot 10^{4} mm \\ \hline 1,66\cdot 10^{7} \mu m \\ \hline 0,0166 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 18 m \\ \hline 180 dm \\ \hline 1,8\cdot 10^{3} cm \\ \hline 1,8\cdot 10^{4} mm \\ \hline 1,8\cdot 10^{7} \mu m \\ \hline 0,018 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline M=\\ \hline 463 m^2 \\ \hline 4,63\cdot 10^{4} dm^2 \\ \hline 4,63\cdot 10^{6} cm^2 \\ \hline 4,63\cdot 10^{8} mm^2 \\ \hline 4,63 a \\ \hline 0,0463 ha \\ \hline 0,000463 km^2 \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline G=\\ \hline 196 m^2 \\ \hline 1,96\cdot 10^{4} dm^2 \\ \hline 1,96\cdot 10^{6} cm^2 \\ \hline 1,96\cdot 10^{8} mm^2 \\ \hline 1,96 a \\ \hline 0,0196 ha \\ \hline 0,000196 km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 659 m^3 \\ \hline 6,59\cdot 10^{5} dm^3 \\ \hline 6,59\cdot 10^{8} cm^3 \\ \hline 6,59\cdot 10^{11} mm^3 \\ \hline 6,59\cdot 10^{5} l \\ \hline 6,59\cdot 10^{3} hl \\ \hline 6,59\cdot 10^{8} ml \\ \hline \end{array}$