formula.expert

Ground Distance on a Slope

Also known as slope distance · ground distance from map distance · slope distance correction · true distance on a hill

g=m2+r2g = \sqrt{m^{2} + r^{2}}

Worked example: 400 m map, 300 m rise → 500 m on the slope (3-4-5) — press Try an example to run it live, then adjust anything.

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See also
Naismith's Rule (Hiking Time) — The fair way to charge for a hill — in time, which the slope costs plenty of, not distance, which it barely touches.Distance from Pace Count — Paced distances follow the ground, so this is the figure your count actually measures.Percent Grade from Rise and Run — The same rise and run stated as a percent grade, the way the map's road signs would.

Learning zone

A map is a view from directly above, so it shows the horizontal leg of every hill; your boots walk the hypotenuse. Pythagoras settles the difference: ground = √(map² + rise²). The surprise is how forgiving the cosine is — a 20 % grade stretches the walked distance only about 2 % past the map figure, and even a 30 % grade under 5 %. If the map says 3.0 km, the ground says 3.06 on a serious hill.

Which is exactly the point Naismith's rule is built on: the slope barely touches the DISTANCE, so charging hills by extra kilometres would miss the cost almost entirely. The hill's real price is paid in climb, charged separately in time. Legs feel the rise; the tape barely sees it.

Ground Distance on a Slope
g=m2+r2g = \sqrt{m^{2} + r^{2}}
Where
  • gg= Ground distance (m)
  • mm= Map distance (m)
  • rr= Rise (m)
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