Sorry in advance for lacking the appropriate terminology, please help me edit it in below.
Take thease basic shapes:
triangle, square, ..., octigon
pyramid, cube
simplex, hypercube
Each flat surface of a cube is a square. The same applies to others in the rest of the table above.
{circle, sphere, hypersphere} also have a great deal in common with thease series.
Therefore, is it true to say that a surface of a sphere either is, or is in some respects, or can be thought of as being a circle, or more accurately a disc?
My thoughts as to how this could be:
A segment or slice isn't a surface.
Thanks Qiaochu Yuan, in topologically two discs can be used to form a sphere. Although they are not flat discs.
Perhaps as the number of sides approaches infinity, the shape of each side aproaches a circle, although it's apparent that circular surfaces do not fit together to form a sphere.
Certainly it is possible to think of a sphere as being glued together from several disks (in mathematics "circle" refers to the object $x^2 + y^2 = 1$ rather than the object $x^2 + y^2 \le 1$, which is a disk). In fact it suffices to use two: the upper hemisphere and the lower hemisphere of a sphere are topologically disks, and gluing them together at their boundaries gives a sphere.
The study of topological spaces from this point of view used to be known as combinatorial topology, but nowadays it is subsumed under algebraic topology.