Democritus of Abdera, the ancient philosopher of Greece, reasoned over two thousand years ago:
If a cut were made through a cone parallel to its base, how should we conceive of the two opposing surfaces which the cut has produced - as equal or as unequal? If they are unequal, that would imply (the lateral side of) a cone is composed of many breaks and protrusions like steps. On the other hand if they are equal, that would imply that two adjacent intersecting planes are equal, which would mean a right cone, being made up of equal rather than unequal circles, must have the same appearance as a cylinder, which is utterly absurd.
– The Presocratics, Philip Wheelwright Editor, 1997, p. 183
So, is a cut made in such a way produce two surfaces that, according to Democritus, must be different in size? If the surfaces are exactly the same size, then why?
Please see the following link of calculating the lateral surface area of a cone for additional information: Surface area of a cone