One.
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Infinitely many. There are infinitely many lines from the apex to the base. Every point on each of these lines is on the surface of the cone.
The apex angle of the cone is the angle that the apex takes. This is measured by the two intersections the cone takes through a perpendicular plane defined by the apex and the center of the circular base.
The apex.
A Cone has a curved base meeting through an apex while. Pyramid has a polygonal base connecting to an apex.
Cone