Stadia reduction

Horizontal distance, vertical difference, and elevation difference from a stadia rod interval and the sight's zenith angle.

Experimental: not yet fully verified. How results are checked

Computing your own values needs JavaScript and WebAssembly. The worked example below was computed when this page was built, and the documentation under "How we got this" applies.

123.315m

The horizontal distance is 123.315 m and the vertical difference 3.229 m.

Vertical difference
Elevation difference

Provenance

Computed by: survey.reduction.stadia 1.0.0, core 0.1.0
Model: Inclined stadia sight: H = K·s·sin²Z + C·sin Z and V = K·s·sin Z·cos Z + C·cos Z (Ghilani & Wolf 2021, stadia); K = 100 and C = 0 by default
Accuracy: Stadia distances are good to about 1 part in 300 to 1 in 1,000, set by reading the rod interval; the arithmetic is exact
Notes: 1 shown with the answer
Cites: Ghilani, C. D., and Wolf, P. R., Pearson, Elementary Surveying: An Introduction to Geomatics

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Your values

Showing an example. Change anything.

Rod interval Upper hair minus lower hair, like 1.234 m · plain numbers mean ft Zenith angleoptional Like 88°30'00" (or give a vertical angle) · plain numbers mean degrees Instrument height HIoptional Like 1.50 m · plain numbers mean ft Middle-hair readingoptional Like 1.60 m · plain numbers mean ft

More options 3

Vertical angleoptional From the horizon, up positive, like 1°30' · plain numbers mean degrees Stadia constant Koptional The multiplier, 100 on most instruments Additive constant Coptional 0 for internal-focusing instruments · plain numbers mean ft

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How we got thisFormula, worked example, sources, and proof

Model: Inclined stadia sight: H = K·s·sin²Z + C·sin Z and V = K·s·sin Z·cos Z + C·cos Z (Ghilani & Wolf 2021, stadia); K = 100 and C = 0 by default

Accuracy: Stadia distances are good to about 1 part in 300 to 1 in 1,000, set by reading the rod interval; the arithmetic is exact

Worked example: A 1.234 m interval at a zenith of 88°30'. Source: Stadia formulas with K = 100, C = 0.

You enter

Instrument height HI: 1.50 m
Rod interval: 1.234 m
Middle-hair reading: 1.60 m
Zenith angle: 88°30'00"

You get

Horizontal distance: 123.315 m
Vertical difference: 3.229 m
Elevation difference: 3.129 m

Review: Not yet independently reviewed by a licensed surveyor.

Last verified: 2026-09-23, 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

2026-09-23, fixed: Survey tools cite Ghilani and Wolf's Elementary Surveying, 16th edition (2021), but their method notes said "Ghilani & Wolf 2018". The notes now give 2021, the edition cited. No results change. Changelog

Checked against: 6 golden test vectors (download the test vectors, each with its source and tolerance). See how results are checked and every source.

Sources

Elementary Surveying: An Introduction to Geomatics, Ghilani, C. D., and Wolf, P. R., Pearson, 16th edition. Chapter 4 (curvature and refraction), chapter 6 (reducing slope distances), and the total-station chapters (two-face zenith observations and index error).

Terms

HI: height of instrument: The height of the instrument's horizontal axis above the station mark, measured with a tape at setup and added when reducing a sight to an elevation difference. Source: Elementary Surveying: An Introduction to Geomatics

Experimental means this tool has not yet met the stable bar: at least 20 golden vectors, differential tests, and an independent worked example.