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: 10
$ \text{Gegeben:}\\ \text{Länge der Seite } \qquad a \qquad [m] \\ \text{Länge der Seite } \qquad b \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{Rechteckige Pyramide}\\ \textbf{Gegeben:} \\ a=\frac{1}{2}m \qquad b=1m \qquad h=2m \\ \\ \textbf{Rechnung:} \\ \text{Pythagoras im} \bigtriangleup ABC \qquad d=\sqrt{a^2+b^2} \\ d=\sqrt{(\frac{1}{2}m)^2+(1m)^2} =1,12m \\ \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}{2}m}{2}\right)^2+(2m)^2} =2,02m \\ \text{Pythagoras im} \bigtriangleup LM_2S \qquad h_2=\sqrt{\left(\dfrac{b}{2}\right)^2+h^2} \\ h_2=\sqrt{\left(\dfrac{1m}{2}\right)^2+(2m)^2} =2,06m \\ \text{Pythagoras im} \bigtriangleup ALS \qquad s=\sqrt{\left(\dfrac{d}{2}\right)^2+h^2} \\ s=\sqrt{\left(\dfrac{1,12m}{2}\right)^2+(2m)^2} =2,08m \\ \text{Mantelfläche} \qquad M= 2 \cdot \dfrac{1}{2} a \cdot h_2 +2 \cdot \dfrac{1}{2} b \cdot h_1 \\ M= 2 \cdot \dfrac{1}{2} \frac{1}{2}m \cdot 2,06m +2 \cdot \dfrac{1}{2} 1m \cdot 2,02m =3,05m^{2} \\ \text{Grundfläche} \qquad G= a\cdot b \\ G= \frac{1}{2}m\cdot 1m=\frac{1}{2}m^{2} \\ \text{Oberfläche} \qquad O= G+M \\ O= \frac{1}{2}m^{2}+3,05m^{2}=3,55m^{3} \\ \text{Volumen} \qquad V= \dfrac{1}{3} a\cdot b \cdot h \\ V= \dfrac{1}{3} \frac{1}{2}m\cdot 1m \cdot 2m =\frac{1}{3}m^{3} \\ \measuredangle CAS \qquad \tan \eta=\frac{h}{\frac{1}{2}d} \\ \tan \eta=\frac{2m}{\frac{1}{2}1,12m} \\ \eta=74,4 ^{\circ}\\ \measuredangle SM_1L \qquad \tan \epsilon=\frac{h}{\frac{1}{2}a} \\ \tan \epsilon=\frac{2m}{\frac{1}{2}\frac{1}{2}m} \\ \epsilon=82,9^{\circ} \\ \measuredangle SM_2L \qquad \tan \mu=\frac{h}{\frac{1}{2}b} \\ \tan \mu=\frac{2m}{\frac{1}{2}1m} \\ \mu=76^{\circ} \\\\\\ \small \begin{array}{|l|} \hline a=\\ \hline \frac{1}{2} m \\ \hline 5 dm \\ \hline 50 cm \\ \hline 500 mm \\ \hline 5\cdot 10^{5} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline b=\\ \hline 1 m \\ \hline 10 dm \\ \hline 100 cm \\ \hline 10^{3} mm \\ \hline 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 \frac{1}{3} m^3 \\ \hline 333\frac{1}{3} dm^3 \\ \hline 333333\frac{1}{3} cm^3 \\ \hline 333333333\frac{1}{3} mm^3 \\ \hline 333\frac{1}{3} l \\ \hline 3\frac{1}{3} hl \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline d=\\ \hline 1,12 m \\ \hline 11,2 dm \\ \hline 112 cm \\ \hline 1,12\cdot 10^{3} mm \\ \hline 1,12\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h1=\\ \hline 2,02 m \\ \hline 20,2 dm \\ \hline 202 cm \\ \hline 2,02\cdot 10^{3} mm \\ \hline 2,02\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 2,06 m \\ \hline 20,6 dm \\ \hline 206 cm \\ \hline 2,06\cdot 10^{3} mm \\ \hline 2,06\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 2,08 m \\ \hline 20,8 dm \\ \hline 208 cm \\ \hline 2,08\cdot 10^{3} mm \\ \hline 2,08\cdot 10^{6} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline M=\\ \hline 3,05 m^2 \\ \hline 305 dm^2 \\ \hline 3,05\cdot 10^{4} cm^2 \\ \hline 3046340\frac{97}{115} mm^2 \\ \hline 0,0305 a \\ \hline 0,000305 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline G=\\ \hline \frac{1}{2} m^2 \\ \hline 50 dm^2 \\ \hline 5\cdot 10^{3} cm^2 \\ \hline 5\cdot 10^{5} mm^2 \\ \hline 0,005 a \\ \hline 5\cdot 10^{-5} ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 3,55 m^3 \\ \hline 3,55\cdot 10^{3} dm^3 \\ \hline 3546340\frac{97}{115} cm^3 \\ \hline 3,55\cdot 10^{9} mm^3 \\ \hline 3,55\cdot 10^{3} l \\ \hline 35,5 hl \\ \hline \end{array}$