Where does the idea of 'area' come from in Geometric Group Theory? The wikipedia article states that this definition was 'inspired' from Riemannian geometry:

Gromov's proof was in large part informed by analogy with filling area functions for compact Riemannian manifolds where the area of a minimal surface bounding a null-homotopic closed curve is bounded in terms of the length of that curve

Could someone explain what the relationship is between a geometric notion of area, and the formal definition [written down for completeness]:

Let $G = \langle S | R \rangle$ be a finitely presented group. Let $w$ be a word in the free group $F(S)$. Then if $w =_G 1$, we can write:

$$ w = \prod_{i=1}^n u_i r_i^{\pm 1} u_i^{-1} \quad u_i \in F(S); r_i \in R $$

The area of $w$ is defined as $\min \{ n : w = \prod_{i=1}^n u_i r_i^{\pm 1}u_i^{-1} \}$

It's intuitively obvious how the length of the perimeter can be related to $|w|$: if we think of the path walked by $w$ on the Cayley graph starting from (say) the identity, the path will be a loop (as $w =_G 1$). The perimeter is the number of edges we need to traverse, which is the length of the word. On the other hand, this definition of area given above is not transparent. It seems to be saying something like:

count the minimum number of 'irreducible' components needed to write $w$ down.

I am unable to see the geometric content of this definition. I would greatly appreciate one, either by analogy, or a direct explanation on the Cayley graph of $G$.