# Relationship btw. surface areas of a cone within a cylinder

let say we have a cone within a cylinder The volumes of cylinder and cone are related by $$\frac{V_{cone}}{V_{cylinder}} = \frac{1}{3}$$

We have also following formulas for surface areas: $$A_{cylinder} = 2\pi r(r+h) \\ A_{cone} = \pi r(r+ \sqrt{r^2 + h^2})$$

with $$r$$ and $$h$$ being the radius and height.

Quastions:

1. I am actually intrested if there is any simple relationship betwenn these sourfaces, like this: $$A_{cone} = \lambda A_{cylinder}$$
2. Also is following correct: If there are some $$(r_0, h_0)$$ which minimize the area of the cylinder, they will respectively minimize the area of the cone?

thanks

1. You already know the formulas for the two areas. By dividing them you see that the ratio is $$\frac{A_{\text{cone}}}{A_{\text{cylinder}}} = \frac{r+\sqrt{r^2+h^2}}{2(r+h)}$$There is no slick reduction that will somehow make that a constant. Just plugging in a few values of $$r$$ and $$h$$ will show you that it varies with respect to each.
2. Technically this is correct, but only because the value of $$r$$ that minimizes $$A_{\text{cylinder}}$$ is $$r = 0$$ (and any $$h$$), which makes the area $$0$$, and of course it also makes $$A_{\text{cone}} = 0$$. I am sure that is not what you are after.
• Proof: The areas are $\ge 0$ so $0$ is the minimum value. $r =0$ makes both areas $0$ QED. – Paul Sinclair Jun 13 '19 at 3:03