Slope distance to horizontal and vertical

Horizontal distance, vertical difference, and elevation difference from a slope distance and zenith angle, with two-face means and the curvature and refraction correction for long sights.

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.

498.097m

The horizontal distance is 498.097 m and the vertical difference 43.578 m.

Vertical difference

Provenance

Computed by: survey.reduction.slope 1.0.0, core 0.1.0
Model: HD = SD·sin Z and VD = SD·cos Z (Ghilani & Wolf 2021, ch. 6); two-face mean Z = (FL + 360° − FR)/2 with index error (FL + FR − 360°)/2; curvature and refraction (1 − k)·HD²/(2R) with R = 6,371,000 m
Accuracy: Exact for the reduction; the curvature and refraction correction carries the uncertainty of k, which varies with the air near the ground
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

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Slope distance Measured along the line of sight, like 500 m · plain numbers mean ft Zenith angleoptional Face left, from straight up, like 85°00'00" · plain numbers mean degrees Face-right zenithoptional The same target read face right, like 274°59'40", for the two-face mean · plain numbers mean degrees Instrument height HIoptional Above the station mark, like 1.55 m · plain numbers mean ft

More options 4

Target height HRoptional Rod or prism height, like 1.80 m · plain numbers mean ft Vertical angleoptional From the horizon, up positive, like 5° (instead of a zenith) · plain numbers mean degrees Correct curvature beyondoptional Apply curvature and refraction to sights longer than this, like 150 m · plain numbers mean ft Refraction coefficient koptional Default 0.13

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

Model: HD = SD·sin Z and VD = SD·cos Z (Ghilani & Wolf 2021, ch. 6); two-face mean Z = (FL + 360° − FR)/2 with index error (FL + FR − 360°)/2; curvature and refraction (1 − k)·HD²/(2R) with R = 6,371,000 m

Accuracy: Exact for the reduction; the curvature and refraction correction carries the uncertainty of k, which varies with the air near the ground

Worked example: 500 m at a zenith of 85°. Source: add-survey-suite scenario: HD ≈ 498.097 m, VD ≈ 43.578 m. It is golden test vector v001, and every build checks the tool still gives its answer within its tolerance.

You enter

Slope distance: 500 m
Zenith angle: 85°00'00"

You get

Horizontal distance: 498.097 m
Vertical difference: 43.578 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: 10 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

DMS: degrees, minutes, seconds: An angle written as whole degrees, minutes (1/60 degree), and seconds (1/3600 degree), like 40°26'46". Source: ISO 6709 Geographic point location
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
HR: height of rod (target height): The height of the prism or target above the point sighted, subtracted 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.