He tells me that we can't place an indefinite article "a" before the "contour's area" part because "area" in this specific context cannot be in its plural form (areas).
Your colleague is wrong in more than one way.
1. "a" does not belong to "contour's area"
Your colleague is arguing that you should parse the sentence as follows:
[a] [contour's area]
That is incorrect. The correct parsing would be:
[a contour]'s [area]
Therefore, the usage of the indefinite article relies on [contour], not [area].
2. Area can be pluralized, even in this context
His argument can easily be disabled by talking about more than one contour:
Two contours' areas.
Area can be pluralized, as an area (= the mathematical value) refers to the area of one object. If there is more than one object, there is more than one area.
There will, however, always be one combined area. But "combined area" and "area" are not freely interchangeable!
To give your colleague a little credit
It would be correct to say that one contour can only have one area. Therefore, in the context of [a contour], it is true that area can logically never be pluralized, even if it is grammatically allowed.
Logical correctness is not the same as grammatical correctness! For example, "The Earth is smaller than an orange" is grammatically correct, but objectively incorrect.
So, it would be correct to say that a single contour cannot have more than one area. But how do we know that we're talking about a single area?
Because the indefinite article "a" is being used.
The only claim that your colleague can make to being correct, inherently relies on the fact that the indefinite article is used to conclusively state that we are talking about a single contour.
In other words, his argument shoots itself in the foot.