Surveying & Geomatics
Intermediate

Traverse Latitude

North-south component of a traverse line from its length and bearing.

Formula

Lat=Lcos⁡(α)Lat = L \cos(\alpha)

Variables

LatLatitude (N-S)
LLine length
\alphaBearing

Example

L=100 m, α=30°: Lat = 86.6 m north

How it works

In traverse surveying, a line can be split into perpendicular parts: one running north-south and the other east-west. The latitude is the north-south projection, so it is found by taking the line length and multiplying by the cosine of its bearing angle, because cosine gives the adjacent side of a right triangle relative to the chosen angle. This works when the bearing is measured from the north-south axis, which is the usual surveying convention.

When to use it

Use this when reducing a measured traverse line into coordinate components for plotting, checking closure, or computing the next station’s northing. It is especially useful in land surveying, route surveying, and any calculation where bearings and distances must be converted into rectangular components.

Worked example, step by step

Given: L = 250 m, α = 40° Step 1: Use the latitude formula Lat = L cos(α) Step 2: Substitute the values Lat = 250 cos(40°) Step 3: Calculate the cosine cos(40°) ≈ 0.7660 Step 4: Multiply Lat = 250 × 0.7660 = 191.5 m Final answer: The latitude is 191.5 m north if the line is in the northward direction.

Common mistakes to avoid

  • Using sine instead of cosine for latitude
  • Forgetting that the sign depends on direction, so southward components should be negative
  • Putting the bearing into the calculator in the wrong angle mode, such as radians instead of degrees

Did You Know?

Summing latitudes and departures around a loop should return to zero — the check for a closed traverse.

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Reviewed & maintained by the FormulaWon editorial teamLast reviewed September 2026