Beispiel Nr: 08
$ \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=7m \qquad b=14m \qquad h=15m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(7m)^2+(14m)^2} =15,7m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{7m}{2}\right)^2+(15m)^2} =15,4m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{14m}{2}\right)^2+(15m)^2} =16,6m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{15,7m}{2}\right)^2+(15m)^2} =16,9m \\ \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} 7m \cdot 16,6m +2 \cdot \dfrac{1}{2} 14m \cdot 15,4m =332m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 7m\cdot 14m=98m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 98m^{2}+332m^{2}=430m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 7m\cdot 14m \cdot 15m =490m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{15m}{\frac{1}{2}15,7m} \\ \eta=62,4 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{15m}{\frac{1}{2}7m} \\ \epsilon=76,9^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{15m}{\frac{1}{2}14m} \\ \mu=65^{\circ} \\ \small \begin{array}{|l|} \hline a=\\ \hline 7 m \\ \hline 70 dm \\ \hline 700 cm \\ \hline 7\cdot 10^{3} mm \\ \hline 7\cdot 10^{6} \mu m \\ \hline 0,007 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline b=\\ \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}\\ \small \begin{array}{|l|} \hline h=\\ \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 V=\\ \hline 490 m^3 \\ \hline 4,9\cdot 10^{5} dm^3 \\ \hline 4,9\cdot 10^{8} cm^3 \\ \hline 4,9\cdot 10^{11} mm^3 \\ \hline 4,9\cdot 10^{5} l \\ \hline 4,9\cdot 10^{3} hl \\ \hline 4,9\cdot 10^{8} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 15,7 m \\ \hline 157 dm \\ \hline 1,57\cdot 10^{3} cm \\ \hline 1,57\cdot 10^{4} mm \\ \hline 1,57\cdot 10^{7} \mu m \\ \hline 0,0157 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 15,4 m \\ \hline 154 dm \\ \hline 1,54\cdot 10^{3} cm \\ \hline 1,54\cdot 10^{4} mm \\ \hline 1,54\cdot 10^{7} \mu m \\ \hline 0,0154 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 16,6 m \\ \hline 166 dm \\ \hline 1,66\cdot 10^{3} cm \\ \hline 1,66\cdot 10^{4} mm \\ \hline 1,66\cdot 10^{7} \mu m \\ \hline 0,0166 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \hline 16,9 m \\ \hline 169 dm \\ \hline 1,69\cdot 10^{3} cm \\ \hline 1,69\cdot 10^{4} mm \\ \hline 1,69\cdot 10^{7} \mu m \\ \hline 0,0169 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 332 m^2 \\ \hline 3,32\cdot 10^{4} dm^2 \\ \hline 3,32\cdot 10^{6} cm^2 \\ \hline 3,32\cdot 10^{8} mm^2 \\ \hline 3,32 a \\ \hline 0,0332 ha \\ \hline 0,000332 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 98 m^2 \\ \hline 9,8\cdot 10^{3} dm^2 \\ \hline 9,8\cdot 10^{5} cm^2 \\ \hline 9,8\cdot 10^{7} mm^2 \\ \hline 0,98 a \\ \hline 0,0098 ha \\ \hline 9,8\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 430 m^3 \\ \hline 4,3\cdot 10^{5} dm^3 \\ \hline 4,3\cdot 10^{8} cm^3 \\ \hline 4,3\cdot 10^{11} mm^3 \\ \hline 4,3\cdot 10^{5} l \\ \hline 4,3\cdot 10^{3} hl \\ \hline 4,3\cdot 10^{8} ml \\ \hline \end{array}\\$