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A curve is said to have a finite length if you can assign a distinct number to measure it. This means the curve can be "measured" from one point to another in the same way you might measure the distance of a straight line. However, instead of just measuring in a straight line between two points, the measurement follows the curve's path.
Having a finite length is crucial because it allows mathematicians and scientists to work with curves in real-world applications effectively. Finite length is central to the concept of a "rectifiable curve," which is essential in calculus for integrating over curves.
If a curve can be approximated by straight line segments whose total length approaches a specific finite value, it qualifies as having finite length, thereby making it rectifiable.

To understand polygonal paths, imagine breaking a curve into small straight pieces, like constructing a jagged path with sticks. These straight pieces are known as polygonal paths or line segments.
The idea is to use these paths to "describe" or approximate the curve. The closer the segments get to following the curve without big jumps or gaps, the better the approximation.
Using polygonal paths is a technique to simplify complex, wavy curves into manageable, straight-line approximations, making calculation of their properties feasible. By defining the straight segments equating to the curve's direction, it helps in analyzing how long a curve actually is or quantifying how it behaves.

The core condition for a curve to be considered rectifiable is tied to the concept of limits. Think of taking increasingly smaller steps as you approximate the curve more closely with your polygonal paths.
If you add up the lengths of these paths and notice that, as you take more and more of these tiny steps, the total length approaches a specific value, this specific value represents the curve's length.
It is as if these line segments are piecing together the curve little by little, and the closer they fit, the more accurate the overall measurement becomes. Essentially, it's like watching a puzzle build up to showcase the picture of the curve, one piece at a time. If this total approaches a finite number, we conclude that the curve has a finite length and is rectifiable.