Beispiel Nr: 07
$ \text{Gegeben:}\\ \text{Länge der Seite } \qquad a \qquad [m] \\ \text{Länge der Seite } \qquad b \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{Rechteckige Pyramide}\\ \textbf{Gegeben:} \\ a=10m \qquad b=18m \qquad h=9m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(10m)^2+(18m)^2} =20,6m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{10m}{2}\right)^2+(9m)^2} =10,3m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\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{20,6m}{2}\right)^2+(9m)^2} =13,7m \\ \text{Mantelfläche} \qquad M= 2 \cdot \dfrac{1}{2} a \cdot h_2 +2 \cdot \dfrac{1}{2} b \cdot h_1 \\ M= 2 \cdot \dfrac{1}{2} 10m \cdot 12,7m +2 \cdot \dfrac{1}{2} 18m \cdot 10,3m =313m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 10m\cdot 18m=180m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 180m^{2}+313m^{2}=493m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 10m\cdot 18m \cdot 9m =540m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{9m}{\frac{1}{2}20,6m} \\ \eta=41,2 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{9m}{\frac{1}{2}10m} \\ \epsilon=60,9^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{9m}{\frac{1}{2}18m} \\ \mu=45^{\circ} \\ \small \begin{array}{|l|} \hline a=\\ \hline 10 m \\ \hline 100 dm \\ \hline 10^{3} cm \\ \hline 10^{4} mm \\ \hline 10^{7} \mu m \\ \hline 0,01 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline b=\\ \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 540 m^3 \\ \hline 5,4\cdot 10^{5} dm^3 \\ \hline 5,4\cdot 10^{8} cm^3 \\ \hline 5,4\cdot 10^{11} mm^3 \\ \hline 5,4\cdot 10^{5} l \\ \hline 5,4\cdot 10^{3} hl \\ \hline 5,4\cdot 10^{8} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 20,6 m \\ \hline 206 dm \\ \hline 2,06\cdot 10^{3} cm \\ \hline 2,06\cdot 10^{4} mm \\ \hline 2,06\cdot 10^{7} \mu m \\ \hline 0,0206 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 10,3 m \\ \hline 103 dm \\ \hline 1,03\cdot 10^{3} cm \\ \hline 1,03\cdot 10^{4} mm \\ \hline 1,03\cdot 10^{7} \mu m \\ \hline 0,0103 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 13,7 m \\ \hline 137 dm \\ \hline 1,37\cdot 10^{3} cm \\ \hline 1,37\cdot 10^{4} mm \\ \hline 1,37\cdot 10^{7} \mu m \\ \hline 0,0137 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 313 m^2 \\ \hline 3,13\cdot 10^{4} dm^2 \\ \hline 3,13\cdot 10^{6} cm^2 \\ \hline 3,13\cdot 10^{8} mm^2 \\ \hline 3,13 a \\ \hline 0,0313 ha \\ \hline 0,000313 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 180 m^2 \\ \hline 1,8\cdot 10^{4} dm^2 \\ \hline 1,8\cdot 10^{6} cm^2 \\ \hline 1,8\cdot 10^{8} mm^2 \\ \hline 1,8 a \\ \hline 0,018 ha \\ \hline 0,00018 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 493 m^3 \\ \hline 4,93\cdot 10^{5} dm^3 \\ \hline 4,93\cdot 10^{8} cm^3 \\ \hline 4,93\cdot 10^{11} mm^3 \\ \hline 4,93\cdot 10^{5} l \\ \hline 4,93\cdot 10^{3} hl \\ \hline 4,93\cdot 10^{8} ml \\ \hline \end{array}\\$