Beispiel Nr: 11
$\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=4m \qquad b=13m \qquad h=16m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(4m)^2+(13m)^2} =13,6m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{4m}{2}\right)^2+(16m)^2} =16,1m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{13m}{2}\right)^2+(16m)^2} =17,3m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{13,6m}{2}\right)^2+(16m)^2} =17,4m \\ \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} 4m \cdot 17,3m +2 \cdot \dfrac{1}{2} 13m \cdot 16,1m =279m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 4m\cdot 13m=52m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 52m^{2}+279m^{2}=331m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 4m\cdot 13m \cdot 16m =277\frac{1}{3}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{16m}{\frac{1}{2}13,6m} \\ \eta=67 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{16m}{\frac{1}{2}4m} \\ \epsilon=82,9^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{16m}{\frac{1}{2}13m} \\ \mu=67,9^{\circ} \\ \small \begin{array}{|l|} \hline a=\\ \hline 4 m \\ \hline 40 dm \\ \hline 400 cm \\ \hline 4\cdot 10^{3} mm \\ \hline 4\cdot 10^{6} \mu m \\ \hline 0,004 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline b=\\ \hline 13 m \\ \hline 130 dm \\ \hline 1,3\cdot 10^{3} cm \\ \hline 1,3\cdot 10^{4} mm \\ \hline 1,3\cdot 10^{7} \mu m \\ \hline 0,013 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h=\\ \hline 16 m \\ \hline 160 dm \\ \hline 1,6\cdot 10^{3} cm \\ \hline 1,6\cdot 10^{4} mm \\ \hline 1,6\cdot 10^{7} \mu m \\ \hline 0,016 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline V=\\ \hline 277 m^3 \\ \hline 2,77\cdot 10^{5} dm^3 \\ \hline 2,77\cdot 10^{8} cm^3 \\ \hline 2,77\cdot 10^{11} mm^3 \\ \hline 2,77\cdot 10^{5} l \\ \hline 2,77\cdot 10^{3} hl \\ \hline 2,77\cdot 10^{8} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 13,6 m \\ \hline 136 dm \\ \hline 1,36\cdot 10^{3} cm \\ \hline 1,36\cdot 10^{4} mm \\ \hline 1,36\cdot 10^{7} \mu m \\ \hline 0,0136 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 16,1 m \\ \hline 161 dm \\ \hline 1,61\cdot 10^{3} cm \\ \hline 1,61\cdot 10^{4} mm \\ \hline 1,61\cdot 10^{7} \mu m \\ \hline 0,0161 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 17,3 m \\ \hline 173 dm \\ \hline 1,73\cdot 10^{3} cm \\ \hline 1,73\cdot 10^{4} mm \\ \hline 1,73\cdot 10^{7} \mu m \\ \hline 0,0173 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \hline 17,4 m \\ \hline 174 dm \\ \hline 1,74\cdot 10^{3} cm \\ \hline 1,74\cdot 10^{4} mm \\ \hline 1,74\cdot 10^{7} \mu m \\ \hline 0,0174 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 279 m^2 \\ \hline 2,79\cdot 10^{4} dm^2 \\ \hline 2,79\cdot 10^{6} cm^2 \\ \hline 2,79\cdot 10^{8} mm^2 \\ \hline 2,79 a \\ \hline 0,0279 ha \\ \hline 0,000279 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 52 m^2 \\ \hline 5,2\cdot 10^{3} dm^2 \\ \hline 5,2\cdot 10^{5} cm^2 \\ \hline 5,2\cdot 10^{7} mm^2 \\ \hline 0,52 a \\ \hline 0,0052 ha \\ \hline 5,2\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 331 m^3 \\ \hline 3,31\cdot 10^{5} dm^3 \\ \hline 3,31\cdot 10^{8} cm^3 \\ \hline 3,31\cdot 10^{11} mm^3 \\ \hline 3,31\cdot 10^{5} l \\ \hline 3,31\cdot 10^{3} hl \\ \hline 3,31\cdot 10^{8} ml \\ \hline \end{array}\\$