9.2 Horizontal, Slope, and Vertical Distances

9.2 Horizontal, Slope, and Vertical Distances

Quick Answer: Slope distance (SD) is what a tape or electronic distance meter (EDM) measures along the line. Horizontal distance (HD) is the plan length between plumb lines. Vertical distance (VD) is the elevation difference. SD² = HD² + VD². Grade g = VD/HD. If you taped along the slope, HD = SD / √(1+g²) and VD = g × HD. With a total-station zenith angle Z, HD = SD × sin(Z) and VD = SD × cos(Z).

Chapter 6 treated the field habit: do not lay a drawing dimension along the ground. This section is CES Domain III item C — the office reduction of those same three lengths, including percent grade versus slope ratio, and vertical distance from slope and zenith.

The Office Triangle

One measured line has three lengths you must keep separate in the notes:

1. SD — along the tape or EDM beam.
2. HD — between the vertical lines through the endpoints. Stationing, northing/easting, and every plan dimension that is not labeled “slope” are HD.
3. VD — elevation of the far end minus elevation of the near end, along that line. Sign: up is positive if you are reducing a zenith less than 90°; a downhill pointing (Z > 90°) gives a negative VD from the instrument axis.

Pythagoras is the closure check: SD² = HD² + VD². If a computed HD and VD do not rebuild the measured SD, a function was swapped or a grade was applied to the wrong length.

VA is the vertical angle from the horizon. VA = 90° − Z. Then sin(Z) = cos(VA), so the two HD formulas agree. Mixing them — cosine of a zenith, or sine of a horizon VA when you wanted HD — returns VD.

Instrument height and target height belong to trigonometric leveling (Chapter 7). They change the elevation of the ground points; they do not enter the HD formula between vertical axes.

Grade: Rise over Run, Percent, and Ratio

Grade g = rise / run = VD / HD. It is a slope of the horizontal run, not of the slope length.

Percent grade = g × 100. A 4% grade rises 4 ft on 100 ft HD, so g = 0.04.

Ratio on California civil plans is usually horizontal : vertical. A 2:1 slope means 2 ft H to 1 ft V, so g = 1/2 = 0.50 = 50% grade. A 4:1 (H:V) slope is g = 0.25 = 25% grade. A 4% grade is 4 on 100, which is 1 V on 25 H, written 25:1 (H:V).

The exam trap is treating “4:1” as 4% or treating percent grade as a ratio of SD. Percent is always vertical on horizontal unless the problem explicitly says the percentage was applied along the tape.

Worked Example 1 — Tape 215.00 ft along a 4% Grade

A crew tapes SD = 215.00 ft along a uniform 4% grade (g = 0.04). Find HD and the elevation difference.

The hypotenuse factor is √(1 + g²) = √(1 + 0.0016) = √1.0016 = 1.000800.

HD = 215.00 / 1.000800 = 214.83 ft
VD = 0.04 × 214.83 = 8.59 ft

Equivalent trig: VA = arctan(0.04) = 2°17′ (2.2906°).
cos(VA) = 0.999201, sin(VA) = 0.039968.
HD = 215.00 × 0.999201 = 214.83 ft.
VD = 215.00 × 0.039968 = 8.59 ft.

Check: √(214.83² + 8.59²) = √(46,151.9 + 73.8) = √46,225.7 = 215.00 ft.

If the party books 215.00 ft as stationing, they are 0.17 ft long in the plan (215.00 − 214.83). A tempting shortcut, VD ≈ 0.04 × 215.00 = 8.60 ft, is only 0.01 ft high because 4% is gentle; the HD error is the one that moves a hub. The same shortcut on a steep grade is not harmless.

Approximate slope correction (Section 9.3) rebuilds the same HD:
C_h ≈ −h² / (2L) = −(8.59)² / (2 × 215.00) = −73.79 / 430 = −0.172 ft
HD ≈ 215.00 − 0.172 = 214.83 ft.

Worked Example 2 — SD 500.00 ft, Zenith 86°12′

A total station measures SD = 500.00 ft at zenith Z = 86°12′ (86.2000°).

sin(86.2000°) = 0.997801
cos(86.2000°) = 0.066274

HD = 500.00 × 0.997801 = 498.90 ft
VD = 500.00 × 0.066274 = 33.14 ft (target above the instrument axis)

Horizon check: VA = 90°00′ − 86°12′ = 3°48′. HD = 500.00 × cos(3°48′) = 498.90 ft. Same pair.

Check: √(498.90² + 33.14²) = √(248,901 + 1,098) = √249,999 = 500.00 ft (to the hundredth).

Wrong function: SD × cos(Z) = 33.14 ft — that is VD, 465 ft short of the plan length. On a nearly horizontal sight the cosine of Z is a small number, so the swapped result looks like a “short shot” instead of an obviously impossible 500-ft layout. Glance at Z: at 86° you should see HD only a little less than SD, not 33 ft.

Worked Example 3 — Plan HD Known (Invert the Triangle)

The drawing calls for HD = 320.00 ft along a 6% grade. The field needs the elevation difference and the slope length to tape.

VD = 0.06 × 320.00 = 19.20 ft
SD = √(320.00² + 19.20²) = √(102,400 + 368.64) = √102,768.64 = 320.58 ft

Taping 320.00 ft along the slope would fall 0.58 ft short of the design point — the same layout bust as Chapter 6, now written as an office inverse.

Worked Example 4 — Grade from Measured HD and VD; Downhill Zenith

A differential level (or a reduced EDM shot) gives HD = 248.50 ft and VD = 12.40 ft between two hubs.

g = 12.40 / 248.50 = 0.04990 = 4.99% (report 5.0% if the problem wants a percent to one decimal).
SD = √(248.50² + 12.40²) = √(61,752.25 + 153.76) = √61,906.01 = 248.81 ft.

Downhill total-station check, so the sign on VD is explicit: SD = 140.00 ft, Z = 93°40′.

HD = 140.00 × sin(93.6667°) = 140.00 × 0.99795 = 139.71 ft
VD = 140.00 × cos(93.6667°) = 140.00 × (−0.06395) = −8.95 ft

The formula does not flip when you look downhill. sin(Z) stays positive until 180°. The negative cosine is the elevation drop from the instrument axis.

Exam Traps

1. SD booked as HD — 215.00 ft along 4% is only 214.83 ft horizontal.
2. Cosine of zenith for HD — 500 × cos(86°12′) = 33.14 ft, the vertical leg.
3. Percent applied to SD for VD — close at 4%, wrong in principle, worse on steep grades.
4. Ratio versus percent — 4:1 (H:V) is 25% grade, not 4%.
5. HI/HT mixed into HD — heights change elevation, not the horizontal reduction.