Suppose I have a cone with a segment removed, as shown in the following image:
https://i.stack.imgur.com/lIziU.png
I can assume all the cone dimensions are known, and that the plane intersecting the cone cutting it into the two sections is parallel to the axis of the cone. Then, I'd like to calculate the surface area of each of the curved sections (I don't need the surface areas of the flat sections). However, I'm not sure how to go about it, as the radial symmetry has been broken.
Apologies if this has been asked before, but I couldn't find the answer anywhere. Any advice would be greatly appreciated, thank you!
If we look at the cone from a point of view parallel to the base. There is a linear relationship between the slant length and the base radius.
And when we look as some section of the wall of the cone, there is a corresponding projection on the base. And the difference in the area is proportional to this $\frac {\text {slant length}}{\text {radius}} = \sec \theta$ ratio.
One factor of distance has been dilated in a linear way, so area has been dilated by the same factor.
If you know multivariate calculus I can show you that way, too.