Beispiel Nr: 06
$ \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=2\frac{1}{2}m \qquad b=3m \qquad h=5m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(2\frac{1}{2}m)^2+(3m)^2} =3,91m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{2\frac{1}{2}m}{2}\right)^2+(5m)^2} =5,15m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{3m}{2}\right)^2+(5m)^2} =5,22m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{3,91m}{2}\right)^2+(5m)^2} =5,37m \\ \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} 2\frac{1}{2}m \cdot 5,22m +2 \cdot \dfrac{1}{2} 3m \cdot 5,15m =28,5m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 2\frac{1}{2}m\cdot 3m=7\frac{1}{2}m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 7\frac{1}{2}m^{2}+28,5m^{2}=36m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 2\frac{1}{2}m\cdot 3m \cdot 5m =12\frac{1}{2}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{5m}{\frac{1}{2}3,91m} \\ \eta=68,7 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{5m}{\frac{1}{2}2\frac{1}{2}m} \\ \epsilon=76^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{5m}{\frac{1}{2}3m} \\ \mu=73,3^{\circ} \\ \small \begin{array}{|l|} \hline a=\\ \hline 2,5 m \\ \hline 25 dm \\ \hline 250 cm \\ \hline 2,5\cdot 10^{3} mm \\ \hline 2,5\cdot 10^{6} \mu m \\ \hline 0,0025 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline b=\\ \hline 3 m \\ \hline 30 dm \\ \hline 300 cm \\ \hline 3\cdot 10^{3} mm \\ \hline 3\cdot 10^{6} \mu m \\ \hline 0,003 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h=\\ \hline 5 m \\ \hline 50 dm \\ \hline 500 cm \\ \hline 5\cdot 10^{3} mm \\ \hline 5\cdot 10^{6} \mu m \\ \hline 0,005 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline V=\\ \hline 12,5 m^3 \\ \hline 1,25\cdot 10^{4} dm^3 \\ \hline 1,25\cdot 10^{7} cm^3 \\ \hline 1,25\cdot 10^{10} mm^3 \\ \hline 1,25\cdot 10^{4} l \\ \hline 125 hl \\ \hline 1,25\cdot 10^{7} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 3,91 m \\ \hline 39,1 dm \\ \hline 391 cm \\ \hline 3,91\cdot 10^{3} mm \\ \hline 3,91\cdot 10^{6} \mu m \\ \hline 0,00391 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 5,15 m \\ \hline 51,5 dm \\ \hline 515 cm \\ \hline 5,15\cdot 10^{3} mm \\ \hline 5,15\cdot 10^{6} \mu m \\ \hline 0,00515 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 5,22 m \\ \hline 52,2 dm \\ \hline 522 cm \\ \hline 5,22\cdot 10^{3} mm \\ \hline 5,22\cdot 10^{6} \mu m \\ \hline 0,00522 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \hline 5,37 m \\ \hline 53,7 dm \\ \hline 537 cm \\ \hline 5,37\cdot 10^{3} mm \\ \hline 5,37\cdot 10^{6} \mu m \\ \hline 0,00537 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 28,5 m^2 \\ \hline 2,85\cdot 10^{3} dm^2 \\ \hline 2,85\cdot 10^{5} cm^2 \\ \hline 2,85\cdot 10^{7} mm^2 \\ \hline 0,285 a \\ \hline 0,00285 ha \\ \hline 2,85\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 7,5 m^2 \\ \hline 750 dm^2 \\ \hline 7,5\cdot 10^{4} cm^2 \\ \hline 7,5\cdot 10^{6} mm^2 \\ \hline 0,075 a \\ \hline 0,00075 ha \\ \hline 7,5\cdot 10^{-6} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 36 m^3 \\ \hline 3,6\cdot 10^{4} dm^3 \\ \hline 3,6\cdot 10^{7} cm^3 \\ \hline 3,6\cdot 10^{10} mm^3 \\ \hline 3,6\cdot 10^{4} l \\ \hline 360 hl \\ \hline 3,6\cdot 10^{7} ml \\ \hline \end{array}\\$