Why the unit circle in $\mathbf{R^2}$ has one dimension?

Solution 1:

You have to define dimension first, but intuitively the dimension is how many independent directions you can walk along if you were on the surface. On a circle, you can only walk back and forth along the circle, you don't have any other choice, so its dimension is $1$. More precisely, a circle is locally homeomorphic to a line, which has dimension $1$.

Solution 2:

I don't know what definition of dimension you are using, so I cannot give a precise answer. But let me say this: As a general principle, each new independent equation introduced in a system of equations cuts down the dimension of the solution space by $1$. This is true for systems of linear equations by elementary linear algebra, but it's also true for algebraic varieties in general, where the precise statement takes the form of the Krull principal ideal theorem.

Since $\mathbf R^2$ has dimension $2$, and the circle is determined by a single equation, it must have dimension $2-1=1$.