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Annotated Elliptic Conical Frustum

What are the formulae for the volume, surface area, and lateral area (i.e. the surface area without the bases) for the above illustrated elliptic conical frustum? I think I've got the volume figured out. $$V=\frac{\pi}{3}\left[(ab)H-(cd)(H-h)\right]\\\textrm{where}\ H=\frac{\sqrt{ab}h}{\sqrt{ab}-\sqrt{cd}}=\frac{ah}{a-c}=\frac{bh}{b-d}$$ But I can't figure out the surface area or lateral area. I know that the general formula for the surface area of a regular polygonal frustum is: $$A+A'+\frac{P+P'}{2}A_p$$ Where $A$ is the area of the large base, $A'$ is the area of of the small base, $P$ is the perimeter of the large base, $P'$ is the perimeter of the small base, and $A_p$ is the apothem (aka slant height). But I'm unsure how I'd translate this to an elliptic conical frustum.

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I can't figure out the surface area or lateral area.

Hint: What is the surface area of the entire or uncut cone ? What is the surface area of the small cone that's been cut ?

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  • $\begingroup$ The surface area of a circular cone would be the area of the base $\pi r^2$ plus the lateral area $\pi rs$ where $s$ is the slant height. But we're talking about an elliptic conical frustum, so the semi-major and semi-minor axis have to be considered. Also, simply subtracting a cut cone won't do I believe because you'd be subtracting the area of the base of the cut cone and not considering the upper base of the frustum. $\endgroup$ – Anthony Vanover Oct 15 '14 at 17:13
  • $\begingroup$ @Antonius: As far as I can tell from your drawing, the two bases are parallel to one another, and the straight line uniting the centers of the two ellipses is perpendicular on both planes. If this is so, then I can't understand for the life of me what the problem is. $\endgroup$ – Lucian Oct 16 '14 at 0:21
  • $\begingroup$ @Lucrian The area of a circle is $\pi r^2$, so likewise the area of an ellipse is $\pi ab$. So the surface area of the elliptic conical frustum is $\pi ab + \pi cd + LA$ where $LA$ is lateral area. I'm not sure what the formula for lateral area is and so I don't know the formula for the surface area. $\endgroup$ – Anthony Vanover Oct 16 '14 at 1:30
  • $\begingroup$ @Antonius: See elliptic cone. $\endgroup$ – Lucian Oct 16 '14 at 1:33
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It seems I've figured this out. The surface area of any frustum is $A+A'+L_A$ where $A$ is the area of the large base, $A'$ is the area of the small base, and $L_A$ is the lateral area. In the case of a circular conical frustum, the area of the bases are $\pi R^2$ and $\pi r^2$ where $R$ is the radius of the big circle and $r$ is the radius of the small circle. The $L_A$ of a general frustum is said to be $\frac{P+P'}{2}A_p$ where $P$ is the perimeter of the big base, $P'$ is the perimeter of the small base, and $A_p$ is the apothem (or slant height). The perimeter of a circle is $2\pi r$, so the lateral area of a circular conical frustum is thus $\pi(R+r)s\ \textrm{where}\ s=\textrm{slant height}$ (notice the 2 cancels out). Therefore, the total surface area of a circular conical frustum is $\pi R^2+\pi r^2+\pi(R+r)s$. Simplifying this, we get $\pi[s(R+r)+R^2+r^2]$.

For an elliptic conical frustum, the semi-major and semi-minor axes have to be considered. So, the area of our bases are $\pi ab$ and $\pi cd$ respectively. The lateral area of an elliptic conical frustum, as with any frustum, requires that we know the perimeters or the bases. So, what is the perimeter of an ellipse? It turns out there are several equations for this because it's not so simple, many of them approximations. But here's my favorite one which happens to be exact: $$P=\pi(a+b)\sum_{n=0}^\infty\binom{0.5}{n}^2\left [ \frac{(a-b)^2}{(a+b)^2} \right ]^n$$ Therefore, the surface area of an elliptic conical frustum is: $$S_A=\pi ab+\pi cd+\frac{s}{2}\left\{ \pi(a+b)\sum_{n=0}^\infty\binom{0.5}{n}^2\left [ \frac{(a-b)^2}{(a+b)^2} \right ]^n + \pi(c+d)\sum_{n=0}^\infty\binom{0.5}{n}^2\left [ \frac{(c-d)^2}{(c+d)^2} \right ]^n \right\}$$ I'm not sure how to simplify that any further, but this is certainly correct.

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I don't think that the answer provided by Anthony is correct. The problem is that the slant length s is not constant, I found here https://rechneronline.de/pi/elliptic-cone.php an approximating formula for the lateral area of an elliptic cone (not truncated!) $$ A := \frac{1}{2} \pi ( a \sqrt{ b^2 + h^2} + b \sqrt{a^2 + h^2 }) $$ i think it can be further generalized for a truncated cone

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