Beispiel Nr: 09
$\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=18m \qquad h=9m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(18m)^2+(18m)^2} =25,5m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{18m}{2}\right)^2+(9m)^2} =12,7m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{25,5m}{2}\right)^2+(9m)^2} =15,6m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 18m \cdot 12,7m =458m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (18m)^2=324m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 324m^{2}+458m^{2}=782m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (18m)^2 \cdot 9m =972m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{9m}{\frac{1}{2}25,5m} \\ \eta=35,3 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{9m}{\frac{1}{2}18m} \\ \epsilon=45^{\circ} \\ \\ \small \begin{array}{|l|} \hline a=\\ \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}\\ \small \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}\\ \small \begin{array}{|l|} \hline V=\\ \hline 972 m^3 \\ \hline 9,72\cdot 10^{5} dm^3 \\ \hline 9,72\cdot 10^{8} cm^3 \\ \hline 9,72\cdot 10^{11} mm^3 \\ \hline 9,72\cdot 10^{5} l \\ \hline 9,72\cdot 10^{3} hl \\ \hline 9,72\cdot 10^{8} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 25,5 m \\ \hline 255 dm \\ \hline 2,55\cdot 10^{3} cm \\ \hline 2,55\cdot 10^{4} mm \\ \hline 2,55\cdot 10^{7} \mu m \\ \hline 0,0255 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 12,7 m \\ \hline 127 dm \\ \hline 1,27\cdot 10^{3} cm \\ \hline 1,27\cdot 10^{4} mm \\ \hline 1,27\cdot 10^{7} \mu m \\ \hline 0,0127 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 12,7 m \\ \hline 127 dm \\ \hline 1,27\cdot 10^{3} cm \\ \hline 1,27\cdot 10^{4} mm \\ \hline 1,27\cdot 10^{7} \mu m \\ \hline 0,0127 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \hline 15,6 m \\ \hline 156 dm \\ \hline 1,56\cdot 10^{3} cm \\ \hline 1,56\cdot 10^{4} mm \\ \hline 1,56\cdot 10^{7} \mu m \\ \hline 0,0156 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 458 m^2 \\ \hline 4,58\cdot 10^{4} dm^2 \\ \hline 4,58\cdot 10^{6} cm^2 \\ \hline 4,58\cdot 10^{8} mm^2 \\ \hline 4,58 a \\ \hline 0,0458 ha \\ \hline 0,000458 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 324 m^2 \\ \hline 3,24\cdot 10^{4} dm^2 \\ \hline 3,24\cdot 10^{6} cm^2 \\ \hline 3,24\cdot 10^{8} mm^2 \\ \hline 3,24 a \\ \hline 0,0324 ha \\ \hline 0,000324 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 782 m^3 \\ \hline 7,82\cdot 10^{5} dm^3 \\ \hline 7,82\cdot 10^{8} cm^3 \\ \hline 7,82\cdot 10^{11} mm^3 \\ \hline 7,82\cdot 10^{5} l \\ \hline 7,82\cdot 10^{3} hl \\ \hline 7,82\cdot 10^{8} ml \\ \hline \end{array}\\$