Beispiel Nr: 05
$ \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=12m \qquad h=14m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(6m)^2+(12m)^2} =13,4m \\ \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+(14m)^2} =14,3m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{12m}{2}\right)^2+(14m)^2} =15,2m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{13,4m}{2}\right)^2+(14m)^2} =15,5m \\ \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 15,2m +2 \cdot \dfrac{1}{2} 12m \cdot 14,3m =263m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 6m\cdot 12m=72m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 72m^{2}+263m^{2}=335m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 6m\cdot 12m \cdot 14m =336m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{14m}{\frac{1}{2}13,4m} \\ \eta=64,4 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{14m}{\frac{1}{2}6m} \\ \epsilon=77,9^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{14m}{\frac{1}{2}12m} \\ \mu=66,8^{\circ} \\\\\\ \tiny \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} \tiny \begin{array}{|l|} \hline b=\\ \hline 12 m \\ \hline 120 dm \\ \hline 1,2\cdot 10^{3} cm \\ \hline 1,2\cdot 10^{4} mm \\ \hline 1,2\cdot 10^{7} \mu m \\ \hline 0,012 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h=\\ \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} \tiny \begin{array}{|l|} \hline V=\\ \hline 336 m^3 \\ \hline 3,36\cdot 10^{5} dm^3 \\ \hline 3,36\cdot 10^{8} cm^3 \\ \hline 3,36\cdot 10^{11} mm^3 \\ \hline 3,36\cdot 10^{5} l \\ \hline 3,36\cdot 10^{3} hl \\ \hline 3,36\cdot 10^{8} ml \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline d=\\ \hline 13,4 m \\ \hline 134 dm \\ \hline 1,34\cdot 10^{3} cm \\ \hline 1,34\cdot 10^{4} mm \\ \hline 1,34\cdot 10^{7} \mu m \\ \hline 0,0134 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h1=\\ \hline 14,3 m \\ \hline 143 dm \\ \hline 1,43\cdot 10^{3} cm \\ \hline 1,43\cdot 10^{4} mm \\ \hline 1,43\cdot 10^{7} \mu m \\ \hline 0,0143 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 15,2 m \\ \hline 152 dm \\ \hline 1,52\cdot 10^{3} cm \\ \hline 1,52\cdot 10^{4} mm \\ \hline 1,52\cdot 10^{7} \mu m \\ \hline 0,0152 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 15,5 m \\ \hline 155 dm \\ \hline 1,55\cdot 10^{3} cm \\ \hline 1,55\cdot 10^{4} mm \\ \hline 1,55\cdot 10^{7} \mu m \\ \hline 0,0155 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline M=\\ \hline 263 m^2 \\ \hline 2,63\cdot 10^{4} dm^2 \\ \hline 2,63\cdot 10^{6} cm^2 \\ \hline 2,63\cdot 10^{8} mm^2 \\ \hline 2,63 a \\ \hline 0,0263 ha \\ \hline 0,000263 km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline G=\\ \hline 72 m^2 \\ \hline 7,2\cdot 10^{3} dm^2 \\ \hline 7,2\cdot 10^{5} cm^2 \\ \hline 7,2\cdot 10^{7} mm^2 \\ \hline 0,72 a \\ \hline 0,0072 ha \\ \hline 7,2\cdot 10^{-5} km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 335 m^3 \\ \hline 3,35\cdot 10^{5} dm^3 \\ \hline 3,35\cdot 10^{8} cm^3 \\ \hline 3,35\cdot 10^{11} mm^3 \\ \hline 3,35\cdot 10^{5} l \\ \hline 3,35\cdot 10^{3} hl \\ \hline 3,35\cdot 10^{8} ml \\ \hline \end{array}$