What is the volume and surface area of the 1-Sphere? I am reading a post on here that mentioned something about the 1-sphere. I know that a 2-sphere is a circle, and 3-sphere is a volume, but what is this 1-sphere and how do you calculate the volume and surface area of a 1-sphere?
A 2-sphere is a standard sphere, if I understand correctly, because the surface itself is a mapping of a two dimensional space (the plane). The standard way of seeing this is called "stereographic projection" in which the circles sits on top of a place, and lines are drawn from the north pole through the sphere to the plane. This is a correspondence between the points in the plane and the points on the sphere.

The 1-sphere is the image of a function of one-variable, so it is a circle. The typical example is to parameterize it in terms of cosine and sine of a real variable $t$.
For a $0$-dimensional sphere: $V = 1$ and $S = 0$

For a $1$-dimensional sphere: $V = 2R$ and $S = 2$

For a $2$-dimensional sphere: $V = \pi R^2$ and $S = 2\pi R$

For a $3$-dimensional sphere: $V = \frac {4}{3} \pi R^3$ and $S = 4 \pi R^2$ 

Note that the derivative of the volume with respect to $R$ equals the surface area. This is true for spheres of every dimension $N$.