Why is the shortest distance between two points not always the path length?
Quick answer
The shortest distance between two points is the straight-line separation between the start and finish. Path length, often called distance traveled, adds up every part of the route actually taken, including bends. So the two quantities can differ. First identify whether a problem asks for the direct separation or the whole route, then use the matching measurement. [S1]
The question, the details, the next step.
Why can the shortest distance between two points be different from the length of the path between them? I keep thinking “distance” must mean just one thing.
The wording can be confusing because a problem may ask about two different measurements. Straight-line separation goes directly from the starting point to the ending point. Path length is the total length of the route you actually follow, including every bend. OpenStax distinguishes distance traveled, the total route length, from displacement, the straight-line change from initial to final position. [S1]
For a winding walk, the direct line between the endpoints cuts across the bends; it does not trace the route. That is why the direct separation and the length of the walk are not necessarily equal. The direct line is the shortest connection between the same endpoints, while the route can be longer. [S1]
Could you show both measurements on one diagram? I’d like to see how to calculate the route without mixing it up with the straight line.
Here is a made-up example. Treat the labels as given measurements, not as measurements from a real place:
A . . . . . . . . . . . . . . .
| .
| 3 m . straight separation
v .
+-----------------------------B
4 m
actual route: down, then acrossSuppose the route goes from A down a 3 m segment, then across a 4 m segment to B. Its path length is 3 m + 4 m = 7 m, because you add the segments traveled. The dotted line represents the direct separation; it is not one of the route segments. The drawing does not give a measurement for that dotted line, so don’t report one unless the problem supplies enough information to find it.
This is the useful distinction: add the pieces of the route for its total length; use the straight start-to-finish measurement when asked for direct separation or displacement. [S1]
What words in a question should I look for? Sometimes it says “how far,” and that doesn’t seem clear enough.
Look for what the question says the measurement should represent, rather than relying on “how far” alone. Phrases such as total route, distance traveled, or length of the path point to adding the route segments. Phrases such as straight-line distance, direct separation, or change from start to finish point to the straight-line quantity. OpenStax likewise says to use distance for the total path and displacement for the straight-line change, and notes that the wording determines which is wanted. [S1]
If the wording is vague, write a short clarification before calculating: “Do you mean the length of the route taken, or the direct separation between the endpoints?” In a school exercise, check the diagram, units, and surrounding instructions for clues. If there still isn’t enough information, state what is missing instead of silently choosing one interpretation.
Can you give me a quick check I can reuse on homework, plus an example of how to write the answer clearly?
Try this three-step check:
- Name the quantity: Is the question asking for the route traveled or the direct start-to-finish separation?
- Match the information: For a route, identify each segment actually traveled. For direct separation, locate or determine the straight-line measurement.
- Show the reasoning: State which quantity you found and include the units supplied by the problem.
Using the made-up route above, a clear answer would be: “The path length is 7 m: 3 m down plus 4 m across. This is the total route, not the straight-line separation from A to B.”
That final sentence prevents the two answers from being mistaken for each other. The example establishes the route total from its stated segment lengths; it does not establish the dotted line’s measurement. [S1]
A practical checklist
Underline words that name the quantity: route length, distance traveled, direct separation, or displacement.
Mark the start and finish so both measurements refer to the same endpoints.
For a route, list each traveled segment once and add its length.
For straight-line separation, use the direct measurement provided or enough information to determine it.
Write the quantity’s name and units with the result; say if the question lacks enough information.
A few more useful answers
If a route has several bends, do I add every segment?
Yes, if the question asks for the total path length, add the lengths of the segments actually traveled. Don’t include the straight line between the endpoints unless it is part of the route.
Does “how far apart” mean the route length?
Not necessarily. It may mean direct separation, while a question about how far someone walked may mean the route traveled. Check the wording and context; ask for clarification if they do not settle it.
Does this diagram tell me the straight-line distance?
No. It gives assumed lengths for the two route segments, which total 7 m. The direct separation is shown but not measured, so the diagram alone does not establish its value.
Sources
References used for this guide.
- [S1] Relative Motion, Distance, and Displacement openstax.org · Checked