How to Approximate Arc Length Using the Distance Formula

To approximate the arc length of a curve using the distance formula, follow these steps:

1. Divide the curve into small straight line segments: To approximate the arc length of a curve, you can divide it into small straight line segments. The smaller the segments, the more accurate the approximation will be.

2. Calculate the distance between each pair of adjacent points: Using the distance formula, calculate the distance between each pair of adjacent points along the curve. Recall that the distance formula is given by:

  d = sqrt((x2 - x1)^2 + (y2 - y1)^2)

  where (x1, y1) and (x2, y2) are the coordinates of two adjacent points along the curve.

3. Sum up the distances: Add up the distances calculated in step 2 to get an approximation of the arc length of the curve.

4. (Optional) Refine the approximation: To get a more accurate approximation, repeat steps 1-3 with smaller segments. You can continue to refine the approximation until you reach a desired level of accuracy.

Note: This method is only an approximation of the arc length of a curve. For a more precise measurement, you may need to use other methods such as calculus.