TITLE: Intuition behind surface of cone being the sector of a circle?
QUESTION [7 upvotes]: While learning how to calculate the surface area of a cylinder, I saw that one of the methods relies on knowing that if we cut a cone, it is the sector of a circle. As shown in the image below:
$\hskip2in$ 

I wasn't able to intuitively see this without getting scissors and paper and doing it myself. I'm wondering if there's any trick or mathematical method that can easily prove this fact without resorting to a "crafting" approach.

REPLY [2 votes]: Think of a pyramid with a regular polygon as the base (hexagon in the figure).

The net of the lateral surface is made of an assembly of identical triangles, which are inscribed in a circular sector.
If you increase the number of sides, the surface tends to a cone and the net to that sector. The aperture arc of the net has a length equal to the perimeter of the base.