Beispiel Nr: 02
$ \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=2m \qquad h=3m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(2m)^2+(2m)^2} =2,83m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{2m}{2}\right)^2+(3m)^2} =3,16m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{2,83m}{2}\right)^2+(3m)^2} =3,32m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 2m \cdot 3,16m =12,6m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (2m)^2=4m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 4m^{2}+12,6m^{2}=16,6m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (2m)^2 \cdot 3m =4m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{3m}{\frac{1}{2}2,83m} \\ \eta=64,8 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{3m}{\frac{1}{2}2m} \\ \epsilon=71,6^{\circ} \\ \\ \small \begin{array}{|l|} \hline a=\\ \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}\\ \small \begin{array}{|l|} \hline h=\\ \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 V=\\ \hline 4 m^3 \\ \hline 4\cdot 10^{3} dm^3 \\ \hline 4\cdot 10^{6} cm^3 \\ \hline 4\cdot 10^{9} mm^3 \\ \hline 4\cdot 10^{3} l \\ \hline 40 hl \\ \hline 4\cdot 10^{6} ml \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 2,83 m \\ \hline 28,3 dm \\ \hline 283 cm \\ \hline 2,83\cdot 10^{3} mm \\ \hline 2,83\cdot 10^{6} \mu m \\ \hline 0,00283 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \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}\\ \small \begin{array}{|l|} \hline h2=\\ \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}\\ \small \begin{array}{|l|} \hline s=\\ \hline 3,32 m \\ \hline 33,2 dm \\ \hline 332 cm \\ \hline 3,32\cdot 10^{3} mm \\ \hline 3,32\cdot 10^{6} \mu m \\ \hline 0,00332 km \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 12,6 m^2 \\ \hline 1,26\cdot 10^{3} dm^2 \\ \hline 1,26\cdot 10^{5} cm^2 \\ \hline 1,26\cdot 10^{7} mm^2 \\ \hline 0,126 a \\ \hline 0,00126 ha \\ \hline 1,26\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 4 m^2 \\ \hline 400 dm^2 \\ \hline 4\cdot 10^{4} cm^2 \\ \hline 4\cdot 10^{6} mm^2 \\ \hline 0,04 a \\ \hline 0,0004 ha \\ \hline 4\cdot 10^{-6} km^2 \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline O=\\ \hline 16,6 m^3 \\ \hline 1,66\cdot 10^{4} dm^3 \\ \hline 1,66\cdot 10^{7} cm^3 \\ \hline 1,66\cdot 10^{10} mm^3 \\ \hline 1,66\cdot 10^{4} l \\ \hline 166 hl \\ \hline 1,66\cdot 10^{7} ml \\ \hline \end{array}\\$