Beispiel Nr: 01
$ \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=4m \qquad h=5m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(4m)^2+(4m)^2} =5,66m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{4m}{2}\right)^2+(5m)^2} =5,39m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{5,66m}{2}\right)^2+(5m)^2} =5,74m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 4m \cdot 5,39m =43,1m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (4m)^2=16m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 16m^{2}+43,1m^{2}=59,1m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (4m)^2 \cdot 5m =26\frac{2}{3}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{5m}{\frac{1}{2}5,66m} \\ \eta=60,5 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{5m}{\frac{1}{2}4m} \\ \epsilon=68,2^{\circ} \\ \\\\\\ \tiny \begin{array}{|l|} \hline a=\\ \hline 4 m \\ \hline 40 dm \\ \hline 400 cm \\ \hline 4\cdot 10^{3} mm \\ \hline 4\cdot 10^{6} \mu m \\ \hline 0,004 km \\ \hline \end{array} \tiny \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} \tiny \begin{array}{|l|} \hline V=\\ \hline 26,7 m^3 \\ \hline 2,67\cdot 10^{4} dm^3 \\ \hline 2,67\cdot 10^{7} cm^3 \\ \hline 2,67\cdot 10^{10} mm^3 \\ \hline 2,67\cdot 10^{4} l \\ \hline 267 hl \\ \hline 2,67\cdot 10^{7} ml \\ \hline \end{array} \tiny \begin{array}{|l|} \hline d=\\ \hline 5,66 m \\ \hline 56,6 dm \\ \hline 566 cm \\ \hline 5,66\cdot 10^{3} mm \\ \hline 5,66\cdot 10^{6} \mu m \\ \hline 0,00566 km \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline h1=\\ \hline 5,39 m \\ \hline 53,9 dm \\ \hline 539 cm \\ \hline 5,39\cdot 10^{3} mm \\ \hline 5,39\cdot 10^{6} \mu m \\ \hline 0,00539 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline h2=\\ \hline 5,39 m \\ \hline 53,9 dm \\ \hline 539 cm \\ \hline 5,39\cdot 10^{3} mm \\ \hline 5,39\cdot 10^{6} \mu m \\ \hline 0,00539 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline s=\\ \hline 5,74 m \\ \hline 57,4 dm \\ \hline 574 cm \\ \hline 5,74\cdot 10^{3} mm \\ \hline 5,74\cdot 10^{6} \mu m \\ \hline 0,00574 km \\ \hline \end{array} \tiny \begin{array}{|l|} \hline M=\\ \hline 43,1 m^2 \\ \hline 4,31\cdot 10^{3} dm^2 \\ \hline 4,31\cdot 10^{5} cm^2 \\ \hline 4,31\cdot 10^{7} mm^2 \\ \hline 0,431 a \\ \hline 0,00431 ha \\ \hline 4,31\cdot 10^{-5} km^2 \\ \hline \end{array}\\ \tiny \begin{array}{|l|} \hline G=\\ \hline 16 m^2 \\ \hline 1,6\cdot 10^{3} dm^2 \\ \hline 1,6\cdot 10^{5} cm^2 \\ \hline 1,6\cdot 10^{7} mm^2 \\ \hline 0,16 a \\ \hline 0,0016 ha \\ \hline 1,6\cdot 10^{-5} km^2 \\ \hline \end{array} \tiny \begin{array}{|l|} \hline O=\\ \hline 59,1 m^3 \\ \hline 5,91\cdot 10^{4} dm^3 \\ \hline 5,91\cdot 10^{7} cm^3 \\ \hline 5,91\cdot 10^{10} mm^3 \\ \hline 5,91\cdot 10^{4} l \\ \hline 591 hl \\ \hline 5,91\cdot 10^{7} ml \\ \hline \end{array}$