A regular hexagon is formed by cutting a congruent equilateral triangle from each of the three corners of an equilateral triangle with side length $12$. What is the length of each side of the hexagon?
Let $x$ be the side length of each small equilateral triangle cut off. Each new cut edge has length $x$, while the remaining segment along each original side has length $12-2x$.
Since the remaining figure is a regular hexagon, these lengths are equal: $x=12-2x$, so $x=4$.
