Geometrie-Stereometrie-Kegelstumpf

• $Kegelstumpf$
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Beispiel Nr: 01
$\text{Gegeben:}\\\text{Körperhöhe} \qquad h \qquad [m] \\ \text{Kreiszahl} \qquad \pi \qquad [] \\ \text{Radius 2} \qquad r_{2} \qquad [m] \\ \text{Radius 1} \qquad r_{1} \qquad [m] \\ \\ \text{Gesucht:} \\\text{Volumen} \qquad V \qquad [m^{3}] \\ \text{Oberfläche} \qquad O \qquad [m^{2}] \\ \\ Kegelstumpf\\ \textbf{Gegeben:} \\ h=6m \qquad \pi=3\frac{16}{113} \qquad r_{2}=2m \qquad r_{1}=4m \\ \\ \textbf{Rechnung:} \\ h=6m\\ \pi=3\frac{16}{113}\\ r_{2}=2m\\ r_{1}=4m\\ h_2=\dfrac{r_2\cdot h}{ r_1-r_2} \\ h_2=\dfrac{2m\cdot 6m}{ 4m-2m}=6m \\ h_1=h_2+h \\ h_1=6m+6m \\ \text{Pythagoras} \\ s_2=\sqrt{r_2^2+h_2^2} \quad s_1=\sqrt{r_1^2+h_1^2} \\ s_2=\sqrt{(2m)^2+(6m)^2}=6,32m \\ s_1=\sqrt{(4m)^2+(12m)^2}=12,6m \\ \text{Mantelfläche} \qquad M= r_1\cdot \pi \cdot s_1-r_2 \cdot \pi \cdot s_2 \\ M= 4m\cdot \pi \cdot 12,6m-2m \cdot \pi \cdot 6,32m=119m^{2} \\ \text{Grund- und Deckfläche} \qquad G= r_1^2\pi \quad D= r_2^2\pi \\ G= (4m)^2\pi=50,3m^{2} \\ D= (2m)^2\pi=12\frac{64}{113}m^{2} \\ \text{Oberfläche} \qquad O= G+D+M \\ O= 50,3m^{2}+12\frac{64}{113}m^{2}+119m^{2}=182m^{2} \\ \text{Volumen} \qquad V= \dfrac{1}{3} r_1^2\cdot \pi \cdot h_1 -\dfrac{1}{3} r_2^2\cdot \pi\cdot h_2 \\ V =\dfrac{1}{3} 4m ^{2} \cdot \pi \cdot 12m - \dfrac{1}{3} 2m ^{2} \cdot \pi\cdot 6m =176m^{3} \\\\\\ \small \begin{array}{|l|} \hline r1=\\ \hline 4 m \\ \hline 40 dm \\ \hline 400 cm \\ \hline 4\cdot 10^{3} mm \\ \hline 4\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline r2=\\ \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 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 176 m^3 \\ \hline 1,76\cdot 10^{5} dm^3 \\ \hline 175929191\frac{1}{5} cm^3 \\ \hline 1,76\cdot 10^{11} mm^3 \\ \hline 1,76\cdot 10^{5} l \\ \hline 1,76\cdot 10^{3} hl \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 176 m \\ \hline 1,76\cdot 10^{3} dm \\ \hline 1,76\cdot 10^{4} cm \\ \hline 1,76\cdot 10^{5} mm \\ \hline 175929191\frac{1}{5} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h1=\\ \hline 12 m \\ \hline 120 dm \\ \hline 1,2\cdot 10^{3} cm \\ \hline 1,2\cdot 10^{4} mm \\ \hline 1,2\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \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 s1=\\ \hline 12,6 m \\ \hline 126 dm \\ \hline 1,26\cdot 10^{3} cm \\ \hline 1,26\cdot 10^{4} mm \\ \hline 1,26\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s2=\\ \hline 6,32 m \\ \hline 63,2 dm \\ \hline 632 cm \\ \hline 6,32\cdot 10^{3} mm \\ \hline 6,32\cdot 10^{6} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 119 m^2 \\ \hline 1,19\cdot 10^{4} dm^2 \\ \hline 1,19\cdot 10^{6} cm^2 \\ \hline 1,19\cdot 10^{8} mm^2 \\ \hline 1,19 a \\ \hline 0,0119 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline G=\\ \hline 50,3 m^2 \\ \hline 5,03\cdot 10^{3} dm^2 \\ \hline 502654\frac{104}{125} cm^2 \\ \hline 50265483\frac{1}{5} mm^2 \\ \hline 0,503 a \\ \hline 0,00503 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline D=\\ \hline 12\frac{64}{113} m^2 \\ \hline 1,26\cdot 10^{3} dm^2 \\ \hline 1,26\cdot 10^{5} cm^2 \\ \hline 12566370\frac{4}{5} mm^2 \\ \hline 0,126 a \\ \hline 0,00126 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 182 m^2 \\ \hline 1,82\cdot 10^{4} dm^2 \\ \hline 1,82\cdot 10^{6} cm^2 \\ \hline 1,82\cdot 10^{8} mm^2 \\ \hline 1,82 a \\ \hline 0,0182 ha \\ \hline \end{array}$