楕円錐台の体積

楕円錐台の底面と上面の半軸と高さから体積、側面積、表面積を計算します。

体積 V
側面積 F
表面積 S

Reference)
Weisstein, Eric W. "Elliptic Cone." From MathWorld--A Wolfram Web Resource.
http://mathworld.wolfram.com/EllipticCone.html

\(\normalsize Truncated\ elliptic\ cone\\ (0) upper\ semi-axis:\hspace{15px} b_{2} = \frac{b_{1}a_{2}}{a_{1}} \\ (1) volume:\hspace{53px} V= \frac{\pi b_{1}h }{3 a_{1} } ( a_{1}^{2} + a_{1}a_{2} + a_{2}^{2} ) \\ (2) lateral\ area: \\\ \hspace{15px} F= \left ( 2 a_{1} \sqrt{b_{1}^2+ \left( \frac{h a_{1}}{a_{1}-a_{2}} \right)^{2}} - 2 a_{2} \sqrt{ \frac{b_{1}^2 a_{2}^{2}}{a_{1}^{2}} + \left( \frac{h a_{2}}{a_{1}-a_{2}} \right)^{2}} \right ) E \left( \sqrt{ \frac{1- \frac{b_{1}^{2}}{a_{1}^{2}} }{ 1 + \frac{b_{1}^{2} (a_{1}-a_{2})^{2}}{h^2 a_{1}^{2} } } } \right) \\\ \hspace{400px} ( a_{1} \gt b_{1} ) \\\ \\\ \hspace{30px} = \left ( 2 b_{1} \sqrt{a_{1}^2+ \left( \frac{h a_{1}}{a_{1}-a_{2}} \right)^{2}} - \frac{ 2 b_{1} a_{2} }{a_{1}} \sqrt{ a_{2}^{2} + \left( \frac{h a_{2}}{a_{1}-a_{2}} \right)^{2}} \right ) E \left( \sqrt{ \frac{1- \frac{a_{1}^{2}}{b_{1}^{2}} }{ 1 + \frac{(a_{1}-a_{2})^{2}}{h^2} } } \right) \\\ \hspace{400px} ( a_{1} \lt b_{1} ) \\\ \\\ E(k)\ :\ 2{\small nd}\ complete\ elliptic\ integral\\ \\ \\ (3) surface\ area:\hspace{10px} S=F+\pi a_{1} b_{1} + \pi a_{2} b_{2} \\\)

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