We have all folded the two straight parts of a sector together to form a cone, perhaps in elementary school. This cone has a 'flat bottom'.
I was asked by a student why it is the case that we have a flat bottom. I thought about it but from a mathematical perspective, I could not come up with an answer.
Why it is the case that when we fold a sector, we will always get a flat bottom cone?
I think it is better to ask why can we get a plane circle for the base. By construction all points on the circular arc are the same distance from the apex. If you form the base to be a circle, all the points of the arc are the same distance from the center of that circle. At least two of them are at right angles to the line from the apex to the center of the base circle. We can then use the converse of the Pythagorean theorem to say all the points of the base circle are at right angles to the line from the apex to the center. This shows the points are all in a plane. As to why we do get a plane circle, we like symmetry, so we make the cone in a way that gives us a plane circle.