Why do we denote $S^1$ for the the unit circle and $S^2$ for unit sphere?

Question

Maybe a quite easy question. Why is $S^1$ the unit circle and $S^2$ is the unit sphere?

Also why is $S^1\times S^1$ a torus? It does not seem that they have anything in common, do they?

Answer 1

$S$ means sphere and 1 resp. 2 gives the number of free parameters. From Sphere:Generalization to other dimensions.

Spheres can be generalized to spaces of any dimension. For any natural number $n$, an "n-sphere," often written as $S^n$, is the set of points in $(n + 1)$-dimensional Euclidean space that are at a fixed distance $r$ from a central point of that space, where $r$ is, as before, a positive real number.

When you put the center of a unit circle at every point of another unit circle you get a torus.

Answer 2

Probably not going to be the best answer, but $S^n$ is a $n$ dimensional manifold in general. I.e, we can write this out in $\mathbb{R}^{n+1}$ as

$$ x_0^2 + \ldots + x_n^2 =1 $$

I'm sure you've seen how these give a circle and a sphere in the case of $n=1,2$.

For understanding the Cartesian product $S^1 \times S^1$, you can think of it as gluing a circle to every point of another circle. This gives a torus.