# Variation of the chart scale in the Mercator projection

June 20, 2017 SorinMStamate

### The variation of the chart scale in the Mercator projection

In the Mercator projection, the chart scale varies, increasing proportionally with (sec ϕ) from the Equator to the poles.

The length of an arc of parallel of latitude between two meridians is equal to the product of the arc of Equator comprised between the same meridians and the (cos ϕ).

Between two given meridians, the arc of parallel decreases from Equator to the poles, in proportion to (cos ϕ).

In the Mercator projection, the segments of parallels remain constant over the entire surface of the map.

So, they increase relative to the corresponding real size, proportional to (sec ϕ), as the distance to the Equator increases.

This means that the chart scale increases proportionally with (sec ϕ) from the Equator to the poles.

Knowing the scale of the map at the Equator, it can be calculated using the formulas below, the scale for any given parallel.

Also, knowing the scale for a given parallel (e.g. the main parallel) one can calculate the scale for any other parallels, determining initially the scale at the Equator.

The MS Excel file contains various examples that can be deleted and edited with your data.

Variation of the chart scale in the Mercator projection _ v1.0