Geometrie-Stereometrie-Kreiskegel


  • $V = \frac{1}{3}\cdot r^{2} \cdot \pi \cdot h$
    1 2
    $r = \sqrt{\frac{3\cdot V}{\pi \cdot h}}$
    1
    $h = \frac{3\cdot V}{r^{2} \cdot \pi }$
    1
    $O = r\cdot \pi \cdot (r+s)$
    1
    $s = \frac{ O}{r\cdot \pi } - r$
    1 2 3
    $r = \frac{-\pi \cdot s + \sqrt{(\pi \cdot s)^{2} +4\cdot \pi \cdot O}}{ 2\cdot \pi }$
    1 2 3
    $M = r\cdot \pi \cdot s$
    $s = \frac{ M}{r\cdot \pi }$
    1 2 3 4
    $r = \frac{ M}{s\cdot \pi }$
    1 2 3 4 5
    $s =\sqrt{h^{2} + r^{2} }$
    1
    $r =\sqrt{s^{2} - h^{2} }$
    1
    $h =\sqrt{s^{2} - r^{2} }$
    1

Beispiel Nr: 02
$ \text{Gegeben:}\\\text{Oberfläche} \qquad O \qquad [m^{2}] \\ \text{Mantellinie} \qquad s \qquad [m] \\ \text{Kreiszahl} \qquad \pi \qquad [] \\ \\ \text{Gesucht:} \\ \text{Radius} \qquad r \qquad [m] \\ \\ r = \frac{-\pi \cdot s + \sqrt{(\pi \cdot s)^{2} +4\cdot \pi \cdot O}}{ 2\cdot \pi }\\ \textbf{Gegeben:} \\ O=3\frac{2}{5}m^{2} \qquad s=3\frac{16}{113}m \qquad \pi=120 \qquad \\ \\ \textbf{Rechnung:} \\ r = \frac{-\pi \cdot s + \sqrt{(\pi \cdot s)^{2} +4\cdot \pi \cdot O}}{ 2\cdot \pi } \\ O=3\frac{2}{5}m^{2}\\ s=3\frac{16}{113}m\\ \pi=120\\ r = \frac{-120 \cdot 3\frac{16}{113}m + \sqrt{(\pi \cdot s)^{2} +4\cdot \pi \cdot 3\frac{2}{5}m^{2}}}{ 2\cdot \pi }\\\\r=0,00899m \\\\\\ \small \begin{array}{|l|} \hline O=\\ \hline 3\frac{2}{5} m^2 \\ \hline 340 dm^2 \\ \hline 3,4\cdot 10^{4} cm^2 \\ \hline 3,4\cdot 10^{6} mm^2 \\ \hline 0,034 a \\ \hline 0,00034 ha \\ \hline \end{array} \small \begin{array}{|l|} \hline s=\\ \hline 3\frac{16}{113} m \\ \hline 31,4 dm \\ \hline 314 cm \\ \hline 3,14\cdot 10^{3} mm \\ \hline 3141592\frac{7}{10} \mu m \\ \hline \end{array} \small \begin{array}{|l|} \hline r=\\ \hline 0,00899 m \\ \hline 0,0899 dm \\ \hline 0,899 cm \\ \hline 8,99 mm \\ \hline 8,99\cdot 10^{3} \mu m \\ \hline \end{array}$