10 Apr

To compute the height $H$ of the Gengzhi Goblets so that the 13-gon prism has volume 1 cup ≈ 236 cubic centimeters, and the radius of the pentagon goblet so that its maximum cross section is the same as the 13-gon (and hence its volume is exactly one quarter as large, or 59 cc):

Note that we assume that the height of the 13-gon prism will equal the distance from one vertex to the midpoint of the opposite side. Referring to the Wikipedia page on regular polygons for formulas, that is to say that the height will equal the circumradius plus the apothem. Now the volume $V$ of the 13-gon prism is the height times the area of the 13-gon, which is $\frac{13}2r^2\sin(\tau/13)$. (As usual we use $\tau$ for the radian measure of a full circle, namely $2\pi$, as it makes most formulas clearer in terms of fractions of circles.) Combining these two observations, we get that
$V = \frac{13}2r^2\sin(\tau/13)(r + r\cos(\tau/26)) = r^3(\frac{13}2\sin(\tau/13)(1+\cos(\tau/26))).$
Setting $V$ equal to 236 and solving for $r_{13} = r$, we get $r_{13} = \sqrt[3]{472/(13\sin(\tau/13)(1+\cos(\tau/26))} \approx 3.41$cm. The height is then $H = r_{13}(1+\cos(\tau/26)) \approx 6.72$cm.
Finally, to find the radius $r_5$ such that pentagon with circumradius $r_5$ has the same area as the 13-gon with circumradius $r_{13}$, we set the respective area formulas equal and solve: $r_5 = r_{13}\sqrt{13\sin(\tau/13)/5\sin(\tau/5)} \approx 3.84$cm.