Beispiel Nr: 12
$\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=1m \qquad h=11m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(1m)^2+(1m)^2} =1,41m \\ \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+(11m)^2} =11m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{1,41m}{2}\right)^2+(11m)^2} =11m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 1m \cdot 11m =22m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (1m)^2=1m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 1m^{2}+22m^{2}=23m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (1m)^2 \cdot 11m =3\frac{2}{3}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{11m}{\frac{1}{2}1,41m} \\ \eta=86,3 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{11m}{\frac{1}{2}1m} \\ \epsilon=87,4^{\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 h=\\ \hline 11 m \\ \hline 110 dm \\ \hline 1,1\cdot 10^{3} cm \\ \hline 1,1\cdot 10^{4} mm \\ \hline 1,1\cdot 10^{7} \mu m \\ \hline 0,011 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline V=\\ \hline 3,67 m^3 \\ \hline 3,67\cdot 10^{3} dm^3 \\ \hline 3,67\cdot 10^{6} cm^3 \\ \hline 3,67\cdot 10^{9} mm^3 \\ \hline 3,67\cdot 10^{3} l \\ \hline 36,7 hl \\ \hline 3,67\cdot 10^{6} ml \\ \hline \end{array} \tiny \begin{array}{|l|} \hline d=\\ \hline 1,41 m \\ \hline 14,1 dm \\ \hline 141 cm \\ \hline 1,41\cdot 10^{3} mm \\ \hline 1,41\cdot 10^{6} \mu m \\ \hline 0,00141 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline h1=\\ \hline 11 m \\ \hline 110 dm \\ \hline 1,1\cdot 10^{3} cm \\ \hline 1,1\cdot 10^{4} mm \\ \hline 1,1\cdot 10^{7} \mu m \\ \hline 0,011 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 11 m \\ \hline 110 dm \\ \hline 1,1\cdot 10^{3} cm \\ \hline 1,1\cdot 10^{4} mm \\ \hline 1,1\cdot 10^{7} \mu m \\ \hline 0,011 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 11 m \\ \hline 110 dm \\ \hline 1,1\cdot 10^{3} cm \\ \hline 1,1\cdot 10^{4} mm \\ \hline 1,1\cdot 10^{7} \mu m \\ \hline 0,011 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline M=\\ \hline 22 m^2 \\ \hline 2,2\cdot 10^{3} dm^2 \\ \hline 2,2\cdot 10^{5} cm^2 \\ \hline 2,2\cdot 10^{7} mm^2 \\ \hline 0,22 a \\ \hline 0,0022 ha \\ \hline 2,2\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline G=\\ \hline 1 m^2 \\ \hline 100 dm^2 \\ \hline 10^{4} cm^2 \\ \hline 10^{6} mm^2 \\ \hline 0,01 a \\ \hline 0,0001 ha \\ \hline 10^{-6} km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 23 m^3 \\ \hline 2,3\cdot 10^{4} dm^3 \\ \hline 2,3\cdot 10^{7} cm^3 \\ \hline 2,3\cdot 10^{10} mm^3 \\ \hline 2,3\cdot 10^{4} l \\ \hline 230 hl \\ \hline 2,3\cdot 10^{7} ml \\ \hline \end{array}$