State plane (SPCS83) to latitude and longitude

Model: Inverse of the SPCS83 zone projection on GRS 80

Show your work

1. Zone

   the zone whose projection and origin these coordinates belong to

   410,656.04 m E, 125,950.431 m N = Pennsylvania South (3702)

2. Scale factor

   how much that projection stretched distance where this lands

   at 410,656.04 m E, 125,950.431 m N = 0.9999595

3. Latitude

   φ from the zone's projection, run backwards

   125,950.431 m N in Pennsylvania South = 40.446111°

The same steps an agent gets from the MCP server with explain: true.

Accuracy: Round trips within 1e-12° of latitude and longitude.

When to use this: Use this when plan coordinates have to become geographic ones: State Plane easting and northing in any of the 124 zones back to NAD 83 latitude and longitude, with the convergence and scale factor at that point. It is what turns a plan, a plat, or an agency dataset back into coordinates you can map, compare with GPS, or hand to a tool that works in latitude and longitude.

Limitations: The zone and the unit have to match the ones the coordinates were computed in; the same numbers in the wrong zone or in feet rather than meters land far away without looking wrong. The result is NAD 83, not WGS 84, and the difference reaches a meter or more. The convergence it reports is the angle between grid north and true north at that point, which is what a bearing taken from the plan has to be corrected by before it is a true bearing.

Worked example: Pennsylvania South coordinates in US survey feet. Source: Inverse of the add-geodesy-suite Pennsylvania South scenario: about 40.446111°, -79.982222°.

Review: Not yet independently reviewed by a geodesist.

Last verified: 2026-09-18, when a maintainer last confirmed this tool's sources at the issuer. See the sources ledger.

Status: version 1.0.0, core 0.1.0. See this tool in the verification report.

Changes

• 2026-09-18, changed: The first six tools passed the stable verification bar: a derivation, an independent worked example, invariant and differential tests, and at least 20 golden vectors each. Changelog

Checked against: 25 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

• State Plane Coordinate System of 1983, NOAA Manual NOS NGS 5, Stem, J. E., National Geodetic Survey, NOAA Manual NOS NGS 5 (reprinted with corrections). Zone constants (Appendix) and the Lambert and transverse Mercator mapping equations.
• Coordinate Conversions and Transformations including Formulas, IOGP Publication 373-7-2 (Guidance Note 7-2), International Association of Oil & Gas Producers (IOGP), Revised September 2019. Sections 3.2.1.1 (Lambert Conic Conformal 2SP, EPSG 9802) and 3.2.4 (Hotine Oblique Mercator variant A, EPSG 9812). Read free.
• Transverse Mercator with an accuracy of a few nanometers, Karney, C. F. F., Journal of Geodesy, Vol. 85, No. 8. pp. 475-485 (6th-order Krüger series).

Terms

NAD83: North American Datum of 1983

The North American datum on the GRS 80 ellipsoid, fixed to the North American plate. Realizations include NAD 83(2011). Source: NADCON5 datum transformations

NGS: National Geodetic Survey

The NOAA office that defines the US National Spatial Reference System: datums, geoid models, and state plane coordinates. Source: State Plane Coordinate System of 1983 (NOAA Manual NOS NGS 5)

SPCS83: State Plane Coordinate System of 1983

124 zones of conformal projections on NAD 83, drawn so that distortion stays near 1 part in 10,000 within each zone. Source: State Plane Coordinate System of 1983 (NOAA Manual NOS NGS 5)