7.2 Trigonometric Leveling
Key Takeaways
With zenith angle Z (0° at the zenith), vertical distance along the line of sight is VD = SD × cos(Z). Target elevation = elev of the instrument mark + h.i. + SD × cos(Z) − h.t.
The horizon form is equivalent: VA = 90° − Z and VD = SD × sin(VA) = HD × tan(VA), with HD = SD × sin(Z). Mixing zenith and horizon functions swaps VD with HD.
Worked one-way: mark 520.00 ft, h.i. = 5.28 ft, SD = 412.50 ft, Z = 82°15′30″, h.t. = 4.85 ft → VD = +55.57 ft → target mark 576.00 ft.
Worked depression: mark 318.75 ft, h.i. = 5.12 ft, SD = 287.40 ft, Z = 96°40′00″, h.t. = 6.00 ft → VD = −33.37 ft → target 284.50 ft. Cosine of Z > 90° is negative; do not drop the sign.
Reciprocal observations (A to B and B to A, then mean the two mark-to-mark differences) cancel most curvature and refraction on long sights. Use a level, not trig heights, when the job is a 0.01 ft pad with room for balanced setups.
7.2 Trigonometric Leveling
Quick Answer: Elev_target = elev_inst + h.i. + SD × cos(Z) − h.t., where Z is the zenith angle. The same vertical distance is SD × sin(VA) or HD × tan(VA) if VA is the vertical angle from the horizon (VA = 90° − Z). Reciprocal observations (both directions) cancel most curvature and refraction. Use trig heights on steep slopes, towers, and canyons; use a level when you can balance short sights and you need hundredths.
CES Domain II item B names trigonometric and differential leveling together. Differential leveling (§7.1) is the default elevation method. Trigonometric leveling is what a total station (or a theodolite plus EDM) does whenever the line of sight is not horizontal: you measure slope distance (SD) and a vertical pointing, reduce vertical distance (VD), then move from mark to mark with two tape heights.
Disambiguate names before you hit the calculator:
- elev_inst — elevation of the station mark under the instrument (hub, monument, TP), not the elevation of the telescope.
- h.i. — measured height from that mark up to the trunnion / horizontal axis. Typical values are 5.0–5.5 ft. This is not leveling HI (elevation of a horizontal line of sight).
- h.t. — height of the target (prism, tape mark, or rod reading) above the target station mark.
- Z — zenith angle, 0° straight up, 90° on the horizon, > 90° looking down. Same convention as Chapter 6.
Zenith Form
Along the line of sight:
VD = SD × cos(Z)
HD = SD × sin(Z) (needed only if you use the tan form below)
Then:
Elev_target = elev_inst + h.i. + VD − h.t.
Elev_target = elev_inst + h.i. + SD × cos(Z) − h.t.
Limits that catch a swapped sine/cosine: Z = 90° → cos(90°) = 0 → VD = 0 (purely horizontal). Z = 0° → VD = SD (looking at the zenith). When Z > 90°, cos(Z) is negative, so VD is negative and the target mark is below the trunnion; keep the sign instead of taking an absolute value and guessing “up or down.”
Horizon Form (Vertical Angle from the Horizon)
Some notes, theodolites, and clinometers store VA with 0° on the horizon, positive up, negative down.
VA = 90° − Z
VD = SD × sin(VA)
VD = HD × tan(VA)
Elev_target = elev_inst + h.i. + SD × sin(VA) − h.t.
sin(VA) = cos(Z) on the same pointing, so both forms agree. Feeding a zenith reading into sin(VA) or a horizon VA into cos(Z) returns something close to HD, which is the classic CES trap.
| Convention | Zero | VD | When you see it |
|---|---|---|---|
| Zenith Z | Straight up | SD × cos(Z) | Total station display |
| Horizon VA | Horizon | SD × sin(VA) = HD × tan(VA) | Theodolite, clinometer, older books |
Worked Example 1 — One-Way, Looking Up
Instrument over mark A, elev_A = 520.00 ft.
h.i. = 5.28 ft. Prism over mark B, h.t. = 4.85 ft.
SD = 412.50 ft. Z = 82°15′30″.
82°15′30″ = 82 + 15/60 + 30/3600 = 82.258333°
cos(82.258333°) = 0.134707
VD = 412.50 × 0.134707 = 55.57 ft (rounded to hundredths)
Horizon check: VA = 90° − 82.258333° = 7.741667° = 7°44′30″.
SD × sin(7.741667°) = 55.57 ft.
HD = 412.50 × sin(82.258333°) = 408.74 ft, and HD × tan(7.741667°) = 55.57 ft.
Elev_B = 520.00 + 5.28 + 55.57 − 4.85 = 576.00 ft.
Stack: 520.00 + 5.28 = 525.28 (elevation of the trunnion). 525.28 + 55.57 = 580.85 (elevation of the prism). 580.85 − 4.85 = 576.00 (mark B). Forgetting h.t. leaves you 4.85 ft high; forgetting h.i. leaves you 5.28 ft low. Using SD × sin(Z) as VD would give HD = 408.74 and a nonsense elevation near 929 ft.
At 412 ft, combined curvature and refraction is only about 0.0035 ft — smaller than a hundredth. Reciprocal observations are not what saves this short hillside shot; h.i., h.t., and the correct cosine are.
Worked Example 2 — One-Way, Looking Down (Z > 90°)
Instrument over mark C, elev_C = 318.75 ft. h.i. = 5.12 ft.
Prism over a streambed hub, h.t. = 6.00 ft. SD = 287.40 ft. Z = 96°40′00″ = 96.666667°.
cos(96.666667°) = −0.116092
VD = 287.40 × (−0.116092) = −33.37 ft
Elev_hub = 318.75 + 5.12 + (−33.37) − 6.00 = 284.50 ft.
Equivalent VA = 90° − 96.666667° = −6.666667° (depression). SD × sin(−6.666667°) = −33.37 ft. Taking the absolute value of VD and adding it because “the river is downhill” would put the hub at 318.75 + 5.12 + 33.37 − 6.00 = 351.24 ft — above the instrument, the wrong valley.
Worked Example 3 — Reciprocal Observations (Brief)
C+R and unmodeled refraction grow with distance squared. Across a canyon you cannot balance 40-ft level sights, so you occupy both ends.
Marks A and B, elev_A = 210.00 ft, SD = 3,850 ft, h.i. = h.t. = 5.40 ft (heights cancel in each one-way Δh).
A → B: Z = 88°12′00″.
VD_AB = 3,850 × cos(88.20°) = +120.93 ft.
One-way elev_B = 210.00 + 120.93 = 330.93 ft.
B → A (same day, similar atmosphere): Z = 91°48′20″.
VD_BA = 3,850 × cos(91.805556°) = −121.30 ft.
That is the height of A minus the height of B from B’s pointing, so the implied Δh_AB = +121.30 ft.
One-way elev_B from this pointing = 210.00 + 121.30 = 331.30 ft.
Mean Δh_AB = (120.93 + 121.30) / 2 = 121.115 → 121.12 ft.
Elev_B (reciprocal) = 210.00 + 121.12 = 331.12 ft.
The two one-ways differ by 0.37 ft. Combined C+R at 3,850 ft is about 0.31 ft (Chapter 9 develops 0.574 M² with M in miles: 3,850/5,280 ≈ 0.729 mi). Reciprocal averaging removes most of that systematic bend; a single long one-way does not. Heights that do not cancel (different h.i. and h.t. on the two occupations) stay in each one-way reduction and must still be taped carefully.
When Trig Leveling versus a Level
| Situation | Prefer |
|---|---|
| Pad, invert, or finished floor to 0.01 ft, with room to turn | Automatic or digital level, balanced BS/FS |
| Steep hillside where a level would need a dozen short setups | Trig heights from a total station |
| Top of a tower, bridge soffit, overhead pipe | Trig, or an inverted rod on a level if the sight is short and nearly horizontal |
| River or canyon with no turning points | Reciprocal trig heights |
| Dense topo shots on a surface | Total-station trig or GNSS RTK (§7.3), not a level loop to every shot |
A 3-minute vertical circle used as a “level replacement” over a 200 ft pad shot is the wrong tool; that geometry is in §7.3. Trig leveling is not sloppy leveling — it is a different measurement model. It still needs a checked vertical circle, a prism constant that matches the software, and h.i./h.t. measured to the same hundredth you want in the elevation.
Exam Traps
- Sine on a zenith angle — SD × sin(Z) is HD, not VD.
- Dropping the sign when Z > 90° — cosine is negative; the target is below the trunnion.
- Skipping h.i. or h.t. — a 5 ft tape error is 500 times a 0.01 ft spec.
- Calling leveling HI and total-station h.i. the same number — one is an elevation, the other is a 5-ft offset.
A total station occupies a mark at 520.00 ft. Instrument height h.i. = 5.28 ft, slope distance to the prism = 412.50 ft, zenith angle Z = 82°15′30″, and target height h.t. = 4.85 ft. The elevation of the target mark is closest to which value?
525.28 ft
580.85 ft
570.72 ft
576.00 ft
A mark at 318.75 ft is occupied with h.i. = 5.12 ft. The pointing to a prism (h.t. = 6.00 ft) has SD = 287.40 ft and Z = 96°40′00″. The elevation of the target mark is closest to which value?
284.50 ft
351.24 ft
317.87 ft
290.50 ft
Why occupy both ends of a long canyon sight and mean the two trigonometric height differences?
To double the curvature correction so it can be ignored
To cancel most curvature and refraction, which reverse with the pointing direction
To eliminate the need to measure instrument height and target height
To convert zenith angles into horizontal angles for traverse closure
Sections you finish are checked off in the contents.