real analysis - Homeomorphisms between circles and rectangles

In our topology class we learn that in $\mathbb{R}^2,$ circles and rectangles are homeomorphic to each others.
I can understand the underline idea intuitively.
But can we find an explicit homeomorphic between them?
If so how?

Also our professor said that, "we can describe any point in the rectangle $[0,1]\times[0,1]$ using a single coordinate."
I wonder how such thing is possible.
As I think, for this we need a bijection between $[0,1]\times[0,1]$ and some (closed?) interval in $\mathbb{R}.$
Can some one explain this phenomena?

Answer

One very visual way of seeing such a homeomorphism is to place one inside the other, choose a point that's inside both (say point $P$), and let the image of a point $Q$ in the circle be where the line emanating from $P$ and through $Q$ intersects the rectangle.

EDIT: Added an example image: