Using the following figure of a hollow cone (without surface at the bottom) as reference:
I decided to calculate the surface area using the method of rings. One small ring has area $ 2\pi(l\sin\theta)dl $
$$ \int_0^L 2\pi(l\sin\theta)\, dl = 2\pi(\sin\theta) \int_0^L l\, dl = 2\pi(\sin\theta) \frac{L^2}{2} = \pi(\sin\theta) L^2 $$
As $ \sin\theta $ is $ \frac{R}{L} $:
$$ \pi(\sin\theta) L^2 = \pi R L $$
Which is the right surface area for the cone.
However, I remember that for the surface area of a sphere we can't use this method (slicing the sphere in rings). So I was wondering why does this method work for a cone.
Best Answer
You can do exactly the same way. Let $l$ be the distance from the top of the sphere. Then, using $r$ the distance to the vertical axis, you need to integrate $$\int_0^{\pi R} 2\pi r\ dl$$ So you need to find the relationship between $r$ and $l$. $$r=R\sin\theta=R\sin\frac lR$$ Then the area is $$\int_0^{\pi R}2\pi R\sin\frac lR dl=2\pi R^2\int_0^\pi\sin\theta d\theta=4\pi R^2$$