The grid scale factor is the ratio of a distance on the projection grid to the corresponding distance on the ellipsoid. It exists because a conformal projection cannot preserve distance, and it varies from place to place within a zone because the projection surface departs from the ellipsoid by different amounts in different places.
It is one of the two factors that separate a measured ground distance from a computed grid distance. The other is the elevation factor, and the two are routinely confused. The grid scale factor is purely a projection effect; it has nothing to do with your elevation.
grid scale factor = grid distance ÷ ellipsoid distanceWhere the number comes from
- Grid scale factor (k)
- The ratio of grid distance to ellipsoid distance at a point. On a conformal projection it is the same in every direction at that point, which is why angles survive and only lengths need correcting.
The projection surface — a cone or a cylinder — cuts through the ellipsoid along one or two lines. Along those lines the two surfaces coincide, so k is exactly 1.0. Inside them the projection surface lies below the ellipsoid and grid distances come out short, so k is less than 1. Outside them the projection surface lies above the ellipsoid and grid distances come out long, so k exceeds 1.
In a Lambert conformal conic zone this means k is a function of latitude alone. In a transverse Mercator zone it is a function of easting alone. In a UTM zone the same transverse Mercator rule applies, with a central meridian value fixed at 0.9996.
How much it varies
The State Plane Coordinate System was designed to hold the grid scale factor within about 1 part in 10 000 — that is, between roughly 0.9999 and 1.0001. Expressed in a way a field crew can feel, 1 part in 10 000 is about 0.53 ft per mile, or a tenth of a foot in a thousand feet.
UTM is far coarser. Its central meridian scale factor of 0.9996 is a fixed −400 parts per million by itself, and toward the edge of a zone it climbs above 1.0009. The table below traces the variation across a UTM zone as an illustration of how a transverse Mercator behaves.
| Distance x from central meridian | Grid scale factor k | Departure from 1 in ppm |
|---|---|---|
| 0 m | 0.9996000 | −400.0 |
| 50 000 m | 0.9996308 | −369.2 |
| 100 000 m | 0.9997231 | −276.9 |
| 150 000 m | 0.9998770 | −123.0 |
| 200 000 m | 1.0000924 | +92.4 |
| 255 000 m | 1.0004004 | +400.4 |
| 334 000 m (zone edge at the equator) | 1.0009729 | +972.9 |
Two things are worth reading out of that table. First, k passes through 1.0 somewhere around 180 km from the central meridian — those are the secant lines. Second, the total swing across the zone is nearly 1400 ppm, which is about 1.4 ft per thousand feet of relative difference between a project on the central meridian and one near the edge.
Getting the factor for your project
Modern software computes k from the projection parameters and your position, and that is the right way to obtain it. For a small site, one value at the center of the project is adequate. For anything longer, the honest approach is to compute k at each end of a line and average them, since k varies continuously across the site.
k_line ≈ (k_1 + 4 k_mid + k_2) ÷ 6The refinement matters only on long lines. Over a typical construction site, the difference between the simple mean and the weighted mean is well below a millimeter and not worth the keystrokes.
The mistakes people actually make
- Confusing the grid scale factor with the combined factor. The grid scale factor alone does not account for elevation, and using it as though it did leaves the elevation error in your distances.
- Applying the factor in the wrong direction. Grid equals ground times combined factor; ground equals grid divided by combined factor.
- Using one project-wide value on a long corridor in a zone where scale varies quickly.
- Assuming the factor is negligible because it is close to 1. A factor of 0.99988 is 120 ppm, which is 1.2 ft on a 10 000 ft line — a systematic error that never averages out.
- Applying a linear scale factor to an area. Area scales as the square of the linear factor.
- Forgetting that angles need no correction at all on a conformal projection. Only distances do.
The last point is worth stating positively, because it is the good news in this subject. On any conformal projection — Lambert, transverse Mercator, oblique Mercator, UTM — a measured angle can be used directly on the grid. The arc-to-chord or second-term correction that formally exists is below a second of arc on ordinary lines and is ignored in all but the most precise geodetic work. Distance is the only thing you have to think about.