Geometrie-Stereometrie-Pyramide

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

Beispiel Nr: 06
$\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=15m \qquad h=6m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+a^2} \\ d=\sqrt{(15m)^2+(15m)^2} =21,2m \\ \text{Pythagoras im} \bigtriangleup LM_1S \qquad h_1=\sqrt{\left(\dfrac{a}{2}\right)^2+h^2} \\ h_1=\sqrt{\left(\dfrac{15m}{2}\right)^2+(6m)^2} =9,6m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{21,2m}{2}\right)^2+(6m)^2} =12,2m \\ \text{Mantelfläche} \qquad M= 4 \cdot \dfrac{1}{2} a \cdot h_1 \\ M= 4 \cdot \dfrac{1}{2} 15m \cdot 9,6m =288m^{2} \\ \text{Grundfläche} \qquad G= a^2 \\ G= (15m)^2=225m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= 225m^{2}+288m^{2}=513m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a^2 \cdot h \\ V= \dfrac{1}{3} (15m)^2 \cdot 6m =450m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{6m}{\frac{1}{2}21,2m} \\ \eta=29,5 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{6m}{\frac{1}{2}15m} \\ \epsilon=38,7^{\circ} \\ \\\\\\ \small \begin{array}{|l|} \hline a=\\ \hline 15 m \\ \hline 150 dm \\ \hline 1,5\cdot 10^{3} cm \\ \hline 1,5\cdot 10^{4} mm \\ \hline 1,5\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h=\\ \hline 6 m \\ \hline 60 dm \\ \hline 600 cm \\ \hline 6\cdot 10^{3} mm \\ \hline 6\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline V=\\ \hline 450 m^3 \\ \hline 4,5\cdot 10^{5} dm^3 \\ \hline 4,5\cdot 10^{8} cm^3 \\ \hline 4,5\cdot 10^{11} mm^3 \\ \hline 4,5\cdot 10^{5} l \\ \hline 4,5\cdot 10^{3} hl \\ \hline \end{array} \small \begin{array}{|l|} \hline d=\\ \hline 21,2 m \\ \hline 212 dm \\ \hline 2,12\cdot 10^{3} cm \\ \hline 2,12\cdot 10^{4} mm \\ \hline 2,12\cdot 10^{7} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline h1=\\ \hline 9,6 m \\ \hline 96 dm \\ \hline 960 cm \\ \hline 9,6\cdot 10^{3} mm \\ \hline 9,6\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 9,6 m \\ \hline 96 dm \\ \hline 960 cm \\ \hline 9,6\cdot 10^{3} mm \\ \hline 9,6\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 12,2 m \\ \hline 122 dm \\ \hline 1,22\cdot 10^{3} cm \\ \hline 1,22\cdot 10^{4} mm \\ \hline 1,22\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 288 m^2 \\ \hline 2,88\cdot 10^{4} dm^2 \\ \hline 2,88\cdot 10^{6} cm^2 \\ \hline 2,88\cdot 10^{8} mm^2 \\ \hline 2,88 a \\ \hline 0,0288 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline G=\\ \hline 225 m^2 \\ \hline 2,25\cdot 10^{4} dm^2 \\ \hline 2,25\cdot 10^{6} cm^2 \\ \hline 2,25\cdot 10^{8} mm^2 \\ \hline 2\frac{1}{4} a \\ \hline 0,0225 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 513 m^3 \\ \hline 5,13\cdot 10^{5} dm^3 \\ \hline 5,13\cdot 10^{8} cm^3 \\ \hline 5,13\cdot 10^{11} mm^3 \\ \hline 5,13\cdot 10^{5} l \\ \hline 5,13\cdot 10^{3} hl \\ \hline \end{array}$