Dot to Dot on x-y plane

Starting from the origin on the $x y$ plane I make a straight line to any point with integer coordinates. I then make a straight line from that integer coordinate to another integer coordinate. I continue this for as many lines as I wish until I return the origin again.

Let $a$ be the number of integer coordinates touching the lines that I have drawn.

Let $b$ be the number of integer coordinates enclosed by the shape that I have drawn (not touching the lines).

If $a = 44$ and $b = 61$ , what is the area of the region enclosed by the lines I have drawn?

Note:

• The lines cannot intersect each other.

• The lines cannot return to a coordinate with a line already passing through it, ending or starting on it.

Example:

$(0,0)$ $\rightarrow$ $(0,2)$ $\rightarrow$ $(2,2)$ $\rightarrow$ $(2,0)$ $\rightarrow$ $(0,0)$

$a = 8, b = 1$ Area enclosed by region $=4$