Beispiel Nr: 04
$ \text{Gegeben:}\\ \text{Länge der Seite } \qquad a \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{Quadratische Pyramide}\\ \textbf{Gegeben:} \\ a=5m \qquad h=12m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(5m)^2+(5m)^2} =7,07m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{5m}{2}\right)^2+(12m)^2} =12,3m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{7,07m}{2}\right)^2+(12m)^2} =12,5m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 5m \cdot 12,3m =123m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (5m)^2=25m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 25m^{2}+123m^{2}=148m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (5m)^2 \cdot 12m =100m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{12m}{\frac{1}{2}7,07m} \\ \eta=73,6 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{12m}{\frac{1}{2}5m} \\ \epsilon=78,2^{\circ} \\ \\ \small \begin{array}{|l|} \hline a=\\ \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 h=\\ \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}\\ \small \begin{array}{|l|} \hline V=\\ \hline 100 m^3 \\ \hline 10^{5} dm^3 \\ \hline 10^{8} cm^3 \\ \hline 10^{11} mm^3 \\ \hline 10^{5} l \\ \hline 10^{3} hl \\ \hline 10^{8} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 7,07 m \\ \hline 70,7 dm \\ \hline 707 cm \\ \hline 7,07\cdot 10^{3} mm \\ \hline 7,07\cdot 10^{6} \mu m \\ \hline 0,00707 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 12,3 m \\ \hline 123 dm \\ \hline 1,23\cdot 10^{3} cm \\ \hline 1,23\cdot 10^{4} mm \\ \hline 1,23\cdot 10^{7} \mu m \\ \hline 0,0123 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h2=\\ \hline 12,3 m \\ \hline 123 dm \\ \hline 1,23\cdot 10^{3} cm \\ \hline 1,23\cdot 10^{4} mm \\ \hline 1,23\cdot 10^{7} \mu m \\ \hline 0,0123 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s=\\ \hline 12,5 m \\ \hline 125 dm \\ \hline 1,25\cdot 10^{3} cm \\ \hline 1,25\cdot 10^{4} mm \\ \hline 1,25\cdot 10^{7} \mu m \\ \hline 0,0125 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 123 m^2 \\ \hline 1,23\cdot 10^{4} dm^2 \\ \hline 1,23\cdot 10^{6} cm^2 \\ \hline 1,23\cdot 10^{8} mm^2 \\ \hline 1,23 a \\ \hline 0,0123 ha \\ \hline 0,000123 km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 25 m^2 \\ \hline 2,5\cdot 10^{3} dm^2 \\ \hline 2,5\cdot 10^{5} cm^2 \\ \hline 2,5\cdot 10^{7} mm^2 \\ \hline 0,25 a \\ \hline 0,0025 ha \\ \hline 2,5\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 148 m^3 \\ \hline 1,48\cdot 10^{5} dm^3 \\ \hline 1,48\cdot 10^{8} cm^3 \\ \hline 1,48\cdot 10^{11} mm^3 \\ \hline 1,48\cdot 10^{5} l \\ \hline 1,48\cdot 10^{3} hl \\ \hline 1,48\cdot 10^{8} ml \\ \hline \end{array}\\$