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Reduced Latitude Calculator

Converts geodetic latitude to reduced (parametric) latitude using the ellipsoid flattening.

100% FreeRuns in your browserIntermediate
How to use this calculator

Inputs

Geodetic latitude ⇄ reduced (parametric) latitude β on WGS84, GRS80 or a custom flattening.

Decimal degrees (−31.95), DMS (31 57 0 S) or DDM (31° 57.0′ S).

Ellipsoid

Enter the values and press Calculate.

Or load an example to see a worked result.

Reduced latitude on the auxiliary circle
Not to scale — flattening exaggerated
Reduced latitude on the auxiliary circleQuarter of the ellipse in the meridian plane with the polar axis, the equatorial axis, a surface point, its normal and its radius.Equator (p)Polar axis (Z)φβφ (geodetic)β (reduced)tan β = (1 − f) tan φ
The ellipsoid normal (geodetic φ) and the radius to the centre (geocentric θ) are different directions except at the equator and the poles.

Batch from CSV

One row per calculation, with a header row. Values use the units currently selected in the form above. Up to 10,000 rows; nothing is uploaded.

How the method works

A fixed worked example to explain the method — it is not your result.

Geodetic normal and geocentric radius are different directions
Illustrative method
Geodetic normal and geocentric radius are different directionsQuarter of the ellipse in the meridian plane with the polar axis, the equatorial axis, a surface point, its normal and its radius.Equator (p)Polar axis (Z)φθφ (geodetic)θ (geocentric)Surface: tan θ = (1 − e²) tan φ
The ellipsoid normal (geodetic φ) and the radius to the centre (geocentric θ) are different directions except at the equator and the poles.

Before you rely on this: First-pass guide only. Verify safety-critical or regulated work against the relevant standards, your project requirements and a qualified professional.

How to use this calculator

  1. Choose the direction: geodetic φ → reduced β, or β → φ.
  2. Enter the latitude in decimal degrees, DMS or DDM.
  3. Choose WGS84, GRS80 or a custom flattening (0 is a sphere).
  4. Press Calculate. Read both latitudes and their difference in arcseconds.

How it works

β = atan2((1 − f) sin φ, cos φ), and the reverse φ = atan2(sin β, (1 − f) cos β). The atan2 form is exact at the poles and the equator.

The difference φ − β is taken from the unrounded angles, then shown in arcseconds.

Reduced latitude is the parameter of the ellipsoid's auxiliary sphere used by geodesic methods such as Vincenty's. It is not geocentric latitude (which uses 1 − e², not 1 − f) and it is not a datum change.

Worked example

φ = 45° on WGS84. β = 44.90378785°, so φ − β = 346.3637″. Reversing 44.90378785° returns 45° to better than 1×10⁻¹⁰°.

Common mistakes

  • Using geocentric latitude where reduced latitude is needed. At 45° they differ by about 346″.
  • Assuming the reduced latitude is a new coordinate. It is an intermediate angle inside ellipsoidal formulas.
  • Rounding β before using it in a distance formula — carry the full precision.

Frequently asked questions

When do I need reduced latitude?

It appears in geodesic (Vincenty) distance calculations and some map-projection maths.

Why is the difference largest near 45°?

The correction depends on sin 2φ, so it vanishes at the equator and the poles and peaks in between.

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Cite this tool

Zerdly. (2026). Reduced Latitude Calculator. Retrieved from https://www.zerdly.com/tools/reduced-latitude-calculator/

Tip: Enter any known values to calculate the remaining results.

Designed to run locally in your browser. Tool inputs are not intentionally stored by Zerdly unless stated otherwise.