Geometrie-Stereometrie-Kegelstumpf


  • $Kegelstumpf$
    1 2 3 4

Beispiel Nr: 03
$ \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}=12m \qquad r_{1}=14m \\ \\ \textbf{Rechnung:} \\ h=6m\\ \pi=3\frac{16}{113}\\ r_{2}=12m\\ r_{1}=14m\\ h_2=\dfrac{r_2\cdot h}{ r_1-r_2} \\ h_2=\dfrac{12m\cdot 6m}{ 14m-12m}=36m \\ h_1=h_2+h \\ h_1=36m+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{(12m)^2+(36m)^2}=37,9m \\ s_1=\sqrt{(14m)^2+(42m)^2}=44,3m \\ \text{Mantelfläche} \qquad M= r_1\cdot \pi \cdot s_1-r_2 \cdot \pi \cdot s_2 \\ M= 14m\cdot \pi \cdot 44,3m-12m \cdot \pi \cdot 37,9m=517m^{2} \\ \text{Grund- und Deckfläche} \qquad G= r_1^2\pi \quad D= r_2^2\pi \\ G= (14m)^2\pi=616m^{2} \\ D= (12m)^2\pi=452m^{2} \\ \text{Oberfläche} \qquad O= G+D+M \\ O= 616m^{2}+452m^{2}+517m^{2}=1,58\cdot 10^{3}m^{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} 14m ^{2} \cdot \pi \cdot 42m - \dfrac{1}{3} 12m ^{2} \cdot \pi\cdot 36m =3,19\cdot 10^{3}m^{3} \\\\\\ \small \begin{array}{|l|} \hline r1=\\ \hline 14 m \\ \hline 140 dm \\ \hline 1,4\cdot 10^{3} cm \\ \hline 1,4\cdot 10^{4} mm \\ \hline 1,4\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline r2=\\ \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 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 3,19\cdot 10^{3} m^3 \\ \hline 3,19\cdot 10^{6} dm^3 \\ \hline 3,19\cdot 10^{9} cm^3 \\ \hline 3,19\cdot 10^{12} mm^3 \\ \hline 3,19\cdot 10^{6} l \\ \hline 3,19\cdot 10^{4} hl \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 3,19\cdot 10^{3} m \\ \hline 3,19\cdot 10^{4} dm \\ \hline 3,19\cdot 10^{5} cm \\ \hline 3,19\cdot 10^{6} mm \\ \hline 3,19\cdot 10^{9} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h1=\\ \hline 42 m \\ \hline 420 dm \\ \hline 4,2\cdot 10^{3} cm \\ \hline 4,2\cdot 10^{4} mm \\ \hline 4,2\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline h2=\\ \hline 36 m \\ \hline 360 dm \\ \hline 3,6\cdot 10^{3} cm \\ \hline 3,6\cdot 10^{4} mm \\ \hline 3,6\cdot 10^{7} \mu m \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline s1=\\ \hline 44,3 m \\ \hline 443 dm \\ \hline 4,43\cdot 10^{3} cm \\ \hline 4,43\cdot 10^{4} mm \\ \hline 4,43\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline s2=\\ \hline 37,9 m \\ \hline 379 dm \\ \hline 3,79\cdot 10^{3} cm \\ \hline 3,79\cdot 10^{4} mm \\ \hline 3,79\cdot 10^{7} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline M=\\ \hline 517 m^2 \\ \hline 5,17\cdot 10^{4} dm^2 \\ \hline 5,17\cdot 10^{6} cm^2 \\ \hline 5,17\cdot 10^{8} mm^2 \\ \hline 5,17 a \\ \hline 0,0517 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline G=\\ \hline 616 m^2 \\ \hline 6,16\cdot 10^{4} dm^2 \\ \hline 6,16\cdot 10^{6} cm^2 \\ \hline 615752169\frac{1}{5} mm^2 \\ \hline 6,16 a \\ \hline 0,0616 ha \\ \hline \end{array}\\ \small \begin{array}{|l|} \hline D=\\ \hline 452 m^2 \\ \hline 4,52\cdot 10^{4} dm^2 \\ \hline 4523893\frac{61}{125} cm^2 \\ \hline 452389348\frac{4}{5} mm^2 \\ \hline 4,52 a \\ \hline 0,0452 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline O=\\ \hline 1,58\cdot 10^{3} m^2 \\ \hline 1,58\cdot 10^{5} dm^2 \\ \hline 1,58\cdot 10^{7} cm^2 \\ \hline 1,58\cdot 10^{9} mm^2 \\ \hline 15,8 a \\ \hline 0,158 ha \\ \hline \end{array}$