Volume of n-dim prismoid

Question

As picture below, it is a n-dim prismoid, $S1$ and $S2$ are parallel. If I know the n-1 dim volume of $S1$ and $S2$, and the high $h$ , how to calculate the n-dim volume ?

Answer 1

Consider a hyper-pyramid with height $h$ and base area $S$. Then, if we cut the prismoid at distance $t$ form the top vertex, the section is congruent to the base, hence has volume $(t/h)^{n-1}S$. So, the volume of the pyramid is given by $$\int_0^h \left(\frac th\right)^{n-1}Sdt =\frac{hS}{n}$$

Now, let $h_1, h_2$ be the heights of the small and big pyramids correspondingly. Then $$h_2 - h_1 = h,\text{ and }\,\, \frac{h_1}{h_2} = \frac{S_1^{1/(n-1)}}{S_2^{1/(n-1)}}.$$

So we get $$h_2 = \frac{h\cdot S_2^{1/(n-1)}}{S_2^{1/(n-1)}-S_1^{1/(n-1)}}, h_1 = \frac{h\cdot S_1^{1/(n-1)}}{S_2^{1/(n-1)}-S_1^{1/(n-1)}}$$

Finally, the volume of the prismoid is $$\frac{h_2S_2-h_1S_1}{n}.$$

If we let $s_1 = S_1^{1/(n-1)}$ and $s_2 = S_2^{1/(n-1)}$, then the volume becomes $$\frac{h(s_2^n-s_1^n)}{n(s_2-s_1)} = \frac hn(s_2^{n-1}+s_2^{n-2}s_1+\cdots+s_1^{n-1}).$$ Note we need not only that the bases are parallel but also that they are congruent. Otherwise I don't think there is a definite answer to the question.