Histogram - Key Observation

For each bar, ask: how far can it extend left and right without encountering a shorter bar?

If bar at index $i$ has height $h$, and it can extend from index $l$ to $r$, its maximum rectangle has area $h \times (r - l + 1)$. The answer is the maximum area over all bars.

Finding the extension limits for all bars in $O(n)$ is the challenge. Monotonic stacks do exactly this.