Calculus How To

Length of a Line Segment

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The length of a line segment between two points is calculated with the distance formula:
distance formula for arc length

This formula only works for linear (straight line) functions. If you have a curved function, use the arc length formula instead.

Length of a Line Segment: Example

Example question: What is the length of the line segment for f(x) = 5x + 2 on the interval [0,1]?

Step 1: (Optional) Draw a Graph of the function. This step can help you see where the points you need for the formula are (x1, x2, y1, y2). I used to create this graph:
length of a line segment

A quick look at the line segment between these two points and I’m going to guess that the length should be around 5. That gives me a way to check that my final answer is reasonable.

Step 2: Plug your two (x, y) coordinates into the distance formula.
distance step 3

If you’re having trouble deciding which point goes where, the following graph has the x and y points labeled.
labeled x and y points for distance formula

Step 3: Simplify and solve:

  • (1 – 0)2 = 1
  • (7 – 2)2 = 25
  • 1 + 25 = 26
  • √(26) ≈ 5.1

So my guess at 5 was fairly close.

Length of a Line Segment & Pythagorean Theorem

You might notice that the distance formula is really an application of the Pythagorean theorem. And in fact, you could get to the same answer by using the Pythagorean formula a2 + b2 = c2.
The following image shows that the hypotenuse of the triangle lies on the line segment you’re trying to find the length of:
length of line segment pythagorean

You may be wondering why we use the distance formula to find the length of a line segment instead of the more simple Pythagorean formula. The answer is that while a, b, and c work perfectly well in geometry, it doesn’t translate well to the Euclidean plane of x’s and y’s, especially once you get to variations on the distance formula for curves. You can’t fit a triangle to a curve, but you can fit a series of small lines. This process of fitting small lines to approximate a curve uses integration, and (along with derivatives) it’s one of the two fundamental areas of calculus.

Stephanie Glen. "Length of a Line Segment" From Calculus for the rest of us!

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