Beispiel Nr: 07
$ \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=12m \qquad h=14m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(12m)^2+(12m)^2} =17m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{12m}{2}\right)^2+(14m)^2} =15,2m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{17m}{2}\right)^2+(14m)^2} =16,4m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 12m \cdot 15,2m =366m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (12m)^2=144m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 144m^{2}+366m^{2}=510m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (12m)^2 \cdot 14m =672m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{14m}{\frac{1}{2}17m} \\ \eta=58,8 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{14m}{\frac{1}{2}12m} \\ \epsilon=66,8^{\circ} \\ \\\\\\ \tiny \begin{array}{|l|} \hline a=\\ \hline 12 m \\ \hline 120 dm \\ \hline 1,2\cdot 10^{3} cm \\ \hline 1,2\cdot 10^{4} mm \\ \hline 1,2\cdot 10^{7} \mu m \\ \hline 0,012 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h=\\ \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 V=\\ \hline 672 m^3 \\ \hline 6,72\cdot 10^{5} dm^3 \\ \hline 6,72\cdot 10^{8} cm^3 \\ \hline 6,72\cdot 10^{11} mm^3 \\ \hline 6,72\cdot 10^{5} l \\ \hline 6,72\cdot 10^{3} hl \\ \hline 6,72\cdot 10^{8} ml \\ \hline \end{array} \tiny \begin{array}{|l|} \hline d=\\ \hline 17 m \\ \hline 170 dm \\ \hline 1,7\cdot 10^{3} cm \\ \hline 1,7\cdot 10^{4} mm \\ \hline 1,7\cdot 10^{7} \mu m \\ \hline 0,017 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline h1=\\ \hline 15,2 m \\ \hline 152 dm \\ \hline 1,52\cdot 10^{3} cm \\ \hline 1,52\cdot 10^{4} mm \\ \hline 1,52\cdot 10^{7} \mu m \\ \hline 0,0152 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 15,2 m \\ \hline 152 dm \\ \hline 1,52\cdot 10^{3} cm \\ \hline 1,52\cdot 10^{4} mm \\ \hline 1,52\cdot 10^{7} \mu m \\ \hline 0,0152 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 16,4 m \\ \hline 164 dm \\ \hline 1,64\cdot 10^{3} cm \\ \hline 1,64\cdot 10^{4} mm \\ \hline 1,64\cdot 10^{7} \mu m \\ \hline 0,0164 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline M=\\ \hline 366 m^2 \\ \hline 3,66\cdot 10^{4} dm^2 \\ \hline 3,66\cdot 10^{6} cm^2 \\ \hline 3,66\cdot 10^{8} mm^2 \\ \hline 3,66 a \\ \hline 0,0366 ha \\ \hline 0,000366 km^2 \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline G=\\ \hline 144 m^2 \\ \hline 1,44\cdot 10^{4} dm^2 \\ \hline 1,44\cdot 10^{6} cm^2 \\ \hline 1,44\cdot 10^{8} mm^2 \\ \hline 1,44 a \\ \hline 0,0144 ha \\ \hline 0,000144 km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 510 m^3 \\ \hline 5,1\cdot 10^{5} dm^3 \\ \hline 5,1\cdot 10^{8} cm^3 \\ \hline 5,1\cdot 10^{11} mm^3 \\ \hline 5,1\cdot 10^{5} l \\ \hline 5,1\cdot 10^{3} hl \\ \hline 5,1\cdot 10^{8} ml \\ \hline \end{array}$