---
tag: 7RQ48RV25W
kind: statement
role: proposition
name: Area of a word representing the identity
sources:
  - wikipedia: en:18871460@r1368818220
    locator: Van Kampen diagram, section Dehn functions and isoperimetric functions > Area of a word representing the identity
    note: adopted
    mention: wikipedia://en/18871460#B4W3
    sections: [Statement]
inventory: [8D310DF0ZJ]
---

## Statement

Let *w* ∈ *F*(*A*) be such that *w* = 1 in *G*. Then the *area* of *w*, denoted Area(*w*), is defined as the minimum of the areas of all Van Kampen diagrams with boundary labels *w* (Van Kampen's lemma says that at least one such diagram exists).

One can show that the area of *w* can be equivalently defined as the smallest *n*≥0 such that there exists a representation (♠) expressing *w* as a product in *F*(*A*) of *n* conjugates of the defining relators.