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.
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.