I'm calculating the centroids for a polygon (the city of Morelia in Mexico), but the coordinates that I'm getting are not inside the polygon and are way off from where the polygon is. What could be happening? Are the coordinates from the centroid given in a different forma? (The ones I'm getting are x= 2575360 and y=852065.8)

I used Stata to double check and I'm getting the same result.

Thanks!

A centroid need not be within a polygon, Consider a C or U shaped object we just expect it to be, but mathematically that need not be the case.

Centroid - Wikipedia

If you have an advanced licence you can calculate 'centroid' features one the standard and the other that does some fiddling (label point) that makes the centerish point inside

Feature To Point—Help | ArcGIS for Desktop