# How to calculate grid convergence (True North to Grid North)?

I would like to know how to calculate the grid convergence angle between True North and Grid North for UTM maps.

I am using ArcGIS 10.0 (ArcView licence) and have used the Calculate Grid Convergence Angle tool from the Cartography toolbox; however, I would like to know how to calculate this angle myself (using Excel) to compare against the ArcGIS result.

Also, is there a standalone program or script that could be shared?

• Excel will only be able to compute the grid convergence for projections whose formulas you have code for. A general solution, applicable on any platform that will project geographic coordinates, appears in my reply at gis.stackexchange.com/a/5075/664. – whuber Sep 29 '14 at 15:26

## 2 Answers

The formula for calculating grid convergence (sometimes called meridian convergence) for spherical UTM projections was given (very incorrectly until just now) at How to Calculate North?

In case that is not clear

γ = arctan [tan (λ - λ0) × sin φ]

where

γ is grid convergence,
λ0 is longitude of UTM zone's central meridian,
φ, λ are latitude, longitude of point in question

See Transverse_Mercator_projection#Convergence where it also gives γ as a function of grid coordinates.

• Where do I find λ0 for a given UTM zone? Do you have a link to a list? – evolved Feb 22 '18 at 15:24
• @evolved: Try this jaworski.ca/utmzones.htm or this apsalin.com/utm_central_meridian_lookup.aspx – Martin F Feb 23 '18 at 23:56
• Just curious, what is the difference between this answer and @V. Kelly Bellis answer? The results come out close. – Rex Aug 10 '18 at 14:03
• @Rex: The other seems to be an approximation. – Martin F Aug 10 '18 at 17:48
• I'm surprised I haven't found a formula yet for grid convergence for ellipsoidal UTM projections. AFAIK UTM doesn't typically use a spherical projection. Spherical transverse mercator is common in web maps, however. – Andy Nov 16 '19 at 7:10

I've always used this formula:

CA = (λ - λCM) × sin φ

Given: CA is the grid Convergence Angle for any transverse Mercator projection; λCM is the longitude of the zone's Central Meridian; φ, λ are the latitude and longitude of the point in question.

Pay attention to the resulting algebraic sign!