Beispiel Nr: 09
$ \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=6m \qquad b=8m \qquad h=9m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(6m)^2+(8m)^2} =10m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{6m}{2}\right)^2+(9m)^2} =9,49m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{8m}{2}\right)^2+(9m)^2} =9,85m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{10m}{2}\right)^2+(9m)^2} =10,3m \\ \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} 6m \cdot 9,85m +2 \cdot \dfrac{1}{2} 8m \cdot 9,49m =135m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 6m\cdot 8m=48m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 48m^{2}+135m^{2}=183m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 6m\cdot 8m \cdot 9m =144m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{9m}{\frac{1}{2}10m} \\ \eta=60,9 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{9m}{\frac{1}{2}6m} \\ \epsilon=71,6^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{9m}{\frac{1}{2}8m} \\ \mu=66^{\circ} \\ \small \begin{array}{|l|} \hline a=\\ \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 b=\\ \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 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 144 m^3 \\ \hline 1,44\cdot 10^{5} dm^3 \\ \hline 1,44\cdot 10^{8} cm^3 \\ \hline 1,44\cdot 10^{11} mm^3 \\ \hline 1,44\cdot 10^{5} l \\ \hline 1,44\cdot 10^{3} hl \\ \hline 1,44\cdot 10^{8} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \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 h1=\\ \hline 9,49 m \\ \hline 94,9 dm \\ \hline 949 cm \\ \hline 9,49\cdot 10^{3} mm \\ \hline 9,49\cdot 10^{6} \mu m \\ \hline 0,00949 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 9,85 m \\ \hline 98,5 dm \\ \hline 985 cm \\ \hline 9,85\cdot 10^{3} mm \\ \hline 9,85\cdot 10^{6} \mu m \\ \hline 0,00985 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \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 M=\\ \hline 135 m^2 \\ \hline 1,35\cdot 10^{4} dm^2 \\ \hline 1,35\cdot 10^{6} cm^2 \\ \hline 1,35\cdot 10^{8} mm^2 \\ \hline 1,35 a \\ \hline 0,0135 ha \\ \hline 0,000135 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 48 m^2 \\ \hline 4,8\cdot 10^{3} dm^2 \\ \hline 4,8\cdot 10^{5} cm^2 \\ \hline 4,8\cdot 10^{7} mm^2 \\ \hline 0,48 a \\ \hline 0,0048 ha \\ \hline 4,8\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 183 m^3 \\ \hline 1,83\cdot 10^{5} dm^3 \\ \hline 1,83\cdot 10^{8} cm^3 \\ \hline 1,83\cdot 10^{11} mm^3 \\ \hline 1,83\cdot 10^{5} l \\ \hline 1,83\cdot 10^{3} hl \\ \hline 1,83\cdot 10^{8} ml \\ \hline \end{array}\\$