Height of an inaccessible object

The height of a tower, tree, or building you cannot reach, from zenith angles to its base and top and a horizontal distance, or from two stations in line with it.

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.

47.867m

The object is 47.867 m tall, measured 120 m away.

Top above the instrument
Horizontal distance

Provenance

Computed by: survey.reduction.inaccessible-height 1.0.0, core 0.1.0
Model: One station: height = D·(cot Z_top − cot Z_base). Two stations in line: D = b·cot Z_far / (cot Z_near − cot Z_far) from the near station, then as for one station (Ghilani & Wolf 2021, trigonometric leveling); curvature and refraction (1 − k)·D²/(2R) when D exceeds the threshold
Accuracy: Exact for the geometry; the result carries the angle and distance errors, and assumes the top is plumb over the base
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.

Zenith to the top From the near station, like 70°00'00" · plain numbers mean degrees Zenith to the baseoptional From the same station, like 92°00'00" · plain numbers mean degrees Horizontal distanceoptional Station to the object, like 120 m · plain numbers mean ft

More options 4

Baselineoptional Instead of the distance: from a second station farther away, in line, like 30 m · plain numbers mean ft Zenith to the top from the far stationoptional With the baseline, like 75°00'00" · 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: One station: height = D·(cot Z_top − cot Z_base). Two stations in line: D = b·cot Z_far / (cot Z_near − cot Z_far) from the near station, then as for one station (Ghilani & Wolf 2021, trigonometric leveling); curvature and refraction (1 − k)·D²/(2R) when D exceeds the threshold

Accuracy: Exact for the geometry; the result carries the angle and distance errors, and assumes the top is plumb over the base

Worked example: A tower 120 m away, zenith 70° to the top and 92° to the base. Source: One-station trigonometric height.

You enter

Horizontal distance: 120 m
Zenith to the base: 92°00'00"
Zenith to the top: 70°00'00"

You get

Height: 47.867 m
Top above the instrument: 43.676 m
Horizontal distance: 120 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).

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

Your values

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Changes

Sources

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