Beispiel Nr: 03
$ \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=1m \qquad b=3m \qquad h=2m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(1m)^2+(3m)^2} =3,16m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{1m}{2}\right)^2+(2m)^2} =2,06m \\ \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+(2m)^2} =2\frac{1}{2}m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{3,16m}{2}\right)^2+(2m)^2} =2,55m \\ \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} 1m \cdot 2\frac{1}{2}m +2 \cdot \dfrac{1}{2} 3m \cdot 2,06m =8,68m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= 1m\cdot 3m=3m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 3m^{2}+8,68m^{2}=11,7m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} 1m\cdot 3m \cdot 2m =2m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{2m}{\frac{1}{2}3,16m} \\ \eta=51,7 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{2m}{\frac{1}{2}1m} \\ \epsilon=76^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{2m}{\frac{1}{2}3m} \\ \mu=53,1^{\circ} \\\\\\ \tiny \begin{array}{|l|} \hline a=\\ \hline 1 m \\ \hline 10 dm \\ \hline 100 cm \\ \hline 10^{3} mm \\ \hline 10^{6} \mu m \\ \hline 0,001 km \\ \hline \end{array} \tiny \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} \tiny \begin{array}{|l|} \hline h=\\ \hline 2 m \\ \hline 20 dm \\ \hline 200 cm \\ \hline 2\cdot 10^{3} mm \\ \hline 2\cdot 10^{6} \mu m \\ \hline 0,002 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline V=\\ \hline 2 m^3 \\ \hline 2\cdot 10^{3} dm^3 \\ \hline 2\cdot 10^{6} cm^3 \\ \hline 2\cdot 10^{9} mm^3 \\ \hline 2\cdot 10^{3} l \\ \hline 20 hl \\ \hline 2\cdot 10^{6} ml \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline d=\\ \hline 3,16 m \\ \hline 31,6 dm \\ \hline 316 cm \\ \hline 3,16\cdot 10^{3} mm \\ \hline 3,16\cdot 10^{6} \mu m \\ \hline 0,00316 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h1=\\ \hline 2,06 m \\ \hline 20,6 dm \\ \hline 206 cm \\ \hline 2,06\cdot 10^{3} mm \\ \hline 2,06\cdot 10^{6} \mu m \\ \hline 0,00206 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \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} \tiny \begin{array}{|l|} \hline s=\\ \hline 2,55 m \\ \hline 25,5 dm \\ \hline 255 cm \\ \hline 2,55\cdot 10^{3} mm \\ \hline 2,55\cdot 10^{6} \mu m \\ \hline 0,00255 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline M=\\ \hline 8,68 m^2 \\ \hline 868 dm^2 \\ \hline 8,68\cdot 10^{4} cm^2 \\ \hline 8,68\cdot 10^{6} mm^2 \\ \hline 0,0868 a \\ \hline 0,000868 ha \\ \hline 8,68\cdot 10^{-6} km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline G=\\ \hline 3 m^2 \\ \hline 300 dm^2 \\ \hline 3\cdot 10^{4} cm^2 \\ \hline 3\cdot 10^{6} mm^2 \\ \hline 0,03 a \\ \hline 0,0003 ha \\ \hline 3\cdot 10^{-6} km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 11,7 m^3 \\ \hline 1,17\cdot 10^{4} dm^3 \\ \hline 1,17\cdot 10^{7} cm^3 \\ \hline 1,17\cdot 10^{10} mm^3 \\ \hline 1,17\cdot 10^{4} l \\ \hline 117 hl \\ \hline 1,17\cdot 10^{7} ml \\ \hline \end{array}$