Beispiel Nr: 04
$ \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=8m \qquad b=15m \qquad h=6m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(8m)^2+(15m)^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{8m}{2}\right)^2+(6m)^2} =7,21m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{15m}{2}\right)^2+(6m)^2} =9,6m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{17m}{2}\right)^2+(6m)^2} =10,4m \\ \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} 8m \cdot 9,6m +2 \cdot \dfrac{1}{2} 15m \cdot 7,21m =185m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 8m\cdot 15m=120m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 120m^{2}+185m^{2}=305m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 8m\cdot 15m \cdot 6m =240m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{6m}{\frac{1}{2}17m} \\ \eta=35,2 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{6m}{\frac{1}{2}8m} \\ \epsilon=56,3^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{6m}{\frac{1}{2}15m} \\ \mu=38,7^{\circ} \\ \small \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}\\ \small \begin{array}{|l|} \hline b=\\ \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}\\ \small \begin{array}{|l|} \hline h=\\ \hline 6 m \\ \hline 60 dm \\ \hline 600 cm \\ \hline 6\cdot 10^{3} mm \\ \hline 6\cdot 10^{6} \mu m \\ \hline 0,006 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline V=\\ \hline 240 m^3 \\ \hline 2,4\cdot 10^{5} dm^3 \\ \hline 2,4\cdot 10^{8} cm^3 \\ \hline 2,4\cdot 10^{11} mm^3 \\ \hline 2,4\cdot 10^{5} l \\ \hline 2,4\cdot 10^{3} hl \\ \hline 2,4\cdot 10^{8} ml \\ \hline \end{array}\\ \small \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}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 7,21 m \\ \hline 72,1 dm \\ \hline 721 cm \\ \hline 7,21\cdot 10^{3} mm \\ \hline 7,21\cdot 10^{6} \mu m \\ \hline 0,00721 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 9,6 m \\ \hline 96 dm \\ \hline 960 cm \\ \hline 9,6\cdot 10^{3} mm \\ \hline 9,6\cdot 10^{6} \mu m \\ \hline 0,0096 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \hline 10,4 m \\ \hline 104 dm \\ \hline 1,04\cdot 10^{3} cm \\ \hline 1,04\cdot 10^{4} mm \\ \hline 1,04\cdot 10^{7} \mu m \\ \hline 0,0104 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 185 m^2 \\ \hline 1,85\cdot 10^{4} dm^2 \\ \hline 1,85\cdot 10^{6} cm^2 \\ \hline 1,85\cdot 10^{8} mm^2 \\ \hline 1,85 a \\ \hline 0,0185 ha \\ \hline 0,000185 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 120 m^2 \\ \hline 1,2\cdot 10^{4} dm^2 \\ \hline 1,2\cdot 10^{6} cm^2 \\ \hline 1,2\cdot 10^{8} mm^2 \\ \hline 1,2 a \\ \hline 0,012 ha \\ \hline 0,00012 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 305 m^3 \\ \hline 3,05\cdot 10^{5} dm^3 \\ \hline 3,05\cdot 10^{8} cm^3 \\ \hline 3,05\cdot 10^{11} mm^3 \\ \hline 3,05\cdot 10^{5} l \\ \hline 3,05\cdot 10^{3} hl \\ \hline 3,05\cdot 10^{8} ml \\ \hline \end{array}\\$