Question

pictured here is a sharpened pencil. If we think of it as a composite solid made up of a cylinder and a cone, how many faces does it have

A1
B2
C3
D4

Answer 1

It depends on how you define face given a polyhedron: http://en.wikipedia.org/wiki/Polyhedron

In grade-level schools, flat surfaces are counted as faces, not curved surfaces. (so the answer for the above would be 1 face)

In college we learn that edges of circles are the limits of an infinite number of polygons and surfaces of cylinders, spheres and codes are the limits of their polyhedral counterparts.

Similar to how 1 + 1/2 ... 1/20 has a finite sum but 1 + 1/2 + ... does not, the same is true for curved surfaces. They no longer should be treated with finite edges and faces.

Another argument for stating that the question does not apply to curved surfaces is that the Euler's characteristic ( http://en.wikipedia.org/wiki/Euler_characteristic) should be true for these objects: vertices - edges + faces = 2. When an object like a cylinder or sphere has has no vertices where we can apply the Euler characteristic, I think we should change terminology to reflect cases where surfaces now have no edges, like for spheres or where edges have no vertices, like in cylinders and cones.

It's amazing that 2000 years after Euclid, some issues in geometry are still not settled.