I don't know what the name for this shape is, so in essence it is a cylinder, radius at base $r$, which has had a wedge of the top cut off at an angle so that rather than a circle the upper face is an ellipse. its height at the top of the slanted ellipse is $h_{max}$, and the height at the bottom is $h_{min}$. the volume was easy to calculate, $\pi r^2 \frac{h_{min}+h_{max}}2$. the surface area is harder: the circle is just $\pi r^2$. I am sure that I could calculate the area of the ellipse, I just haven't got round to it, but the area of the once-rectangle is a challenge, as the upper edge is a wave. I assume it involves trigonometry, but I don't know what the formula is. help please? (apologies if my explanation is not clear)
[Math] surface area of ‘cylinder’ with the top cut at an angle
polyhedrasurfaces
Best Answer
sorry, hadn't really thought it through. although it is a wave, it goes above the average height just as much as it goes below it, so it is similar to the volume calculation: $2\pi r \frac{h_{min}+h_{max}}2$. I haven't calculated the surface area of the ellipse yet. EDIT: thanks for ellipse formula. semiminor axis $=r$ and semimajor axis $=\sqrt{r^2+\frac{(h_{max}-h_{min})^2}4}$ area of ellipse $=\pi r\sqrt{r^2+\frac{(h_{max}-h_{min})^2}4}$