Geometrie-Stereometrie-Pyramide


  • $V =\frac{1}{3} G\cdot h$
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    $G = \frac{3 \cdot V}{h}$
    1 2
    $h = \frac{3 \cdot V}{G}$
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    $O = G +M $
    1 2
    $G = O-M$
    1 2 3
    $M = O- G $
    1 2
    $\text{Rechteckige Pyramide}$
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    $\text{Quadratische Pyramide}$
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Beispiel Nr: 03
$ \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=\frac{1}{5}m \qquad h=4\frac{1}{2}m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(\frac{1}{5}m)^2+(\frac{1}{5}m)^2} =0,283m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{\frac{1}{5}m}{2}\right)^2+(4\frac{1}{2}m)^2} =4,5m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{0,283m}{2}\right)^2+(4\frac{1}{2}m)^2} =4,5m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} \frac{1}{5}m \cdot 4,5m =1,8m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (\frac{1}{5}m)^2=\frac{1}{25}m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= \frac{1}{25}m^{2}+1,8m^{2}=1,84m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (\frac{1}{5}m)^2 \cdot 4\frac{1}{2}m =\frac{3}{50}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{4\frac{1}{2}m}{\frac{1}{2}0,283m} \\ \eta=88,2 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{4\frac{1}{2}m}{\frac{1}{2}\frac{1}{5}m} \\ \epsilon=88,7^{\circ} \\ \\\\\\ \small \begin{array}{|l|} \hline a=\\ \hline \frac{1}{5} m \\ \hline 2 dm \\ \hline 20 cm \\ \hline 200 mm \\ \hline 2\cdot 10^{5} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h=\\ \hline 4\frac{1}{2} m \\ \hline 45 dm \\ \hline 450 cm \\ \hline 4,5\cdot 10^{3} mm \\ \hline 4,5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline V=\\ \hline \frac{3}{50} m^3 \\ \hline 60 dm^3 \\ \hline 6\cdot 10^{4} cm^3 \\ \hline 6\cdot 10^{7} mm^3 \\ \hline 60 l \\ \hline \frac{3}{5} hl \\ \hline \end{array} \small \begin{array}{|l|} \hline d=\\ \hline 0,283 m \\ \hline 2,83 dm \\ \hline 28,3 cm \\ \hline 283 mm \\ \hline 2,83\cdot 10^{5} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 4,5 m \\ \hline 45 dm \\ \hline 450 cm \\ \hline 4,5\cdot 10^{3} mm \\ \hline 4,5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 4,5 m \\ \hline 45 dm \\ \hline 450 cm \\ \hline 4,5\cdot 10^{3} mm \\ \hline 4,5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 4,5 m \\ \hline 45 dm \\ \hline 450 cm \\ \hline 4,5\cdot 10^{3} mm \\ \hline 4,5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 1,8 m^2 \\ \hline 180 dm^2 \\ \hline 1,8\cdot 10^{4} cm^2 \\ \hline 1,8\cdot 10^{6} mm^2 \\ \hline 0,018 a \\ \hline 0,00018 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline \frac{1}{25} m^2 \\ \hline 4 dm^2 \\ \hline 400 cm^2 \\ \hline 4\cdot 10^{4} mm^2 \\ \hline 0,0004 a \\ \hline 4\cdot 10^{-6} ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 1,84 m^3 \\ \hline 1,84\cdot 10^{3} dm^3 \\ \hline 1,84\cdot 10^{6} cm^3 \\ \hline 1,84\cdot 10^{9} mm^3 \\ \hline 1,84\cdot 10^{3} l \\ \hline 18,4 hl \\ \hline \end{array}$