Geometrie-Stereometrie-Pyramide

• $V =\frac{1}{3} G\cdot h$
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$G = \frac{3 \cdot V}{h}$
1 2
$h = \frac{3 \cdot V}{G}$
1 2 3
$O = G +M$
1 2
$G = O-M$
1 2 3
$M = O- G$
1 2
$\text{Rechteckige Pyramide}$
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$\text{Quadratische Pyramide}$
1 2 3 4 5 6 7 8 9 10 11 12 13

Beispiel Nr: 05
$\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=3m \qquad h=2m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(3m)^2+(3m)^2} =4,24m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{3m}{2}\right)^2+(2m)^2} =2\frac{1}{2}m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{4,24m}{2}\right)^2+(2m)^2} =2,92m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 3m \cdot 2\frac{1}{2}m =15m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (3m)^2=9m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 9m^{2}+15m^{2}=24m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (3m)^2 \cdot 2m =6m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{2m}{\frac{1}{2}4,24m} \\ \eta=43,3 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{2m}{\frac{1}{2}3m} \\ \epsilon=53,1^{\circ} \\ \\\\\\ \small \begin{array}{|l|} \hline a=\\ \hline 3 m \\ \hline 30 dm \\ \hline 300 cm \\ \hline 3\cdot 10^{3} mm \\ \hline 3\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h=\\ \hline 2 m \\ \hline 20 dm \\ \hline 200 cm \\ \hline 2\cdot 10^{3} mm \\ \hline 2\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline V=\\ \hline 6 m^3 \\ \hline 6\cdot 10^{3} dm^3 \\ \hline 6\cdot 10^{6} cm^3 \\ \hline 6\cdot 10^{9} mm^3 \\ \hline 6\cdot 10^{3} l \\ \hline 60 hl \\ \hline \end{array} \small \begin{array}{|l|} \hline d=\\ \hline 4,24 m \\ \hline 42,4 dm \\ \hline 424 cm \\ \hline 4,24\cdot 10^{3} mm \\ \hline 4,24\cdot 10^{6} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 2\frac{1}{2} m \\ \hline 25 dm \\ \hline 250 cm \\ \hline 2,5\cdot 10^{3} mm \\ \hline 2,5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 2\frac{1}{2} m \\ \hline 25 dm \\ \hline 250 cm \\ \hline 2,5\cdot 10^{3} mm \\ \hline 2,5\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 2,92 m \\ \hline 29,2 dm \\ \hline 292 cm \\ \hline 2,92\cdot 10^{3} mm \\ \hline 2,92\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 15 m^2 \\ \hline 1,5\cdot 10^{3} dm^2 \\ \hline 1,5\cdot 10^{5} cm^2 \\ \hline 1,5\cdot 10^{7} mm^2 \\ \hline \frac{3}{20} a \\ \hline 0,0015 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 9 m^2 \\ \hline 900 dm^2 \\ \hline 9\cdot 10^{4} cm^2 \\ \hline 9\cdot 10^{6} mm^2 \\ \hline \frac{9}{100} a \\ \hline 0,0009 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 24 m^3 \\ \hline 2,4\cdot 10^{4} dm^3 \\ \hline 2,4\cdot 10^{7} cm^3 \\ \hline 2,4\cdot 10^{10} mm^3 \\ \hline 2,4\cdot 10^{4} l \\ \hline 240 hl \\ \hline \end{array}$