---
title: "Tacheometry booking, plotting, gradient A-B"
subject: "Surveying II"
chapter: 2
key: "semester_4/surveying_ii:tacheometry-booking-plotting-gradient-ab-p"
source: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometry-booking-plotting-gradient-ab-p
site: BCT Engineering
---

# Tacheometry booking, plotting, gradient A-B

Surveying II · Chapter 2: Tacheometry

- Asked in: 2073 Bhadra, 4+6 marks
- Video: [Tacheometry Calculation: Table method](https://www.youtube.com/watch?v=aLcB2LJiBDA) — engiNERDing
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tacheometry-booking-plotting-gradient-ab-p
- Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2

## Question

What is tacheometry? Explain the booking and plotting details in tacheometric surveying. Calculate the gradient between station A and station B from the following observations taken from tacheometer fitted with anallatic lens. The RL and HI of instrument station P are 1275 m and 1.55 m respectively.

| Inst. Station | Target Station | Bearing | Vertical angle | Staff readings (m) |
|---|---|---|---|---|
| P | A | 30°30' | 6°30' | 1.115, 1.735, 2.355 |
| | B | 75°30' | 9°15' | 1.250, 2.000, 2.750 |

## Short answer

Tacheometry measures distances and elevations optically with a tacheometer and staff. Book each point with station, HI, bearing, vertical angle, top/axial/bottom readings, s, D, V, RL; plot by bearing and distance, then contours. AB=105.07 m, RL_A=1288.762 m, RL_B=1298.348 m, gradient rising 1 in 11.0 (9.12%).

## Answer

#### Tacheometry
Tacheometry (tachymetry) is a method of surveying in which horizontal distances and differences in elevation are determined from optical measurements with a tacheometer (a theodolite with stadia hairs) and a graduated staff, without chaining or levelling. It is quick and is used for contouring, topographic surveys and traversing over rough, steep or obstructed ground.

#### Booking
Observations are booked in a standard tabular field book, one line for every staff point:

| Inst. stn | HI | Staff stn | Bearing | Vertical angle | Top | Axial | Bottom | s | D | V | RL | Remarks |
|---|---|---|---|---|---|---|---|---|---|---|---|---|

The columns for $s$, $D$, $V$ and RL are filled in during computation. A sketch of the area, with the station names and the position of details, is drawn on the facing page.

#### Plotting
1. Choose a scale and plot the instrument stations (from the control traverse or by coordinates).
2. From each station, mark each staff point by its bearing (with a protractor) and its horizontal distance $D$ at the scale (polar method), or compute its coordinates and plot on the grid.
3. Write the RL beside each point, join details (roads, buildings, streams) using the sketch, interpolate between spot levels and draw contours.
4. Check with the points common to two stations, and finish with the title, scale and north arrow.

#### Numerical: gradient between A and B
Anallatic lens, so $C=0$ and $K=100$ (taken as 100, not stated).

Staff held vertical, line of sight inclined at angle $\theta$ (additive constant zero, which is the case for an anallatic lens or $C=0$):

$$
D=Ks\cos^2\theta,\qquad V=\tfrac12Ks\sin2\theta
$$

where $s$ is the staff intercept, $\theta$ the vertical angle and $K=100$, $C=0$.

#### Step 1: Horizontal distance and vertical component of each sight
**Point A** (bearing 30°30', angle of elevation $6^\circ 30'$):

$$
\begin{aligned}
s&=2.355-1.115=1.240\ \text{m}\\
D&=100\times1.240\times\cos^2 6^\circ 30'=122.411\ \text{m}\\
V&=\tfrac12\times100\times1.240\times\sin 2(6^\circ 30')=13.947\ \text{m}
\end{aligned}
$$

**Point B** (bearing 75°30', angle of elevation $9^\circ 15'$):

$$
\begin{aligned}
s&=2.750-1.250=1.500\ \text{m}\\
D&=100\times1.500\times\cos^2 9^\circ 15'=146.124\ \text{m}\\
V&=\tfrac12\times100\times1.500\times\sin 2(9^\circ 15')=23.798\ \text{m}
\end{aligned}
$$

#### Step 2: Reduced levels
RL of instrument axis = RL of station + HI = 1275.000 + 1.550 = 1276.550 m.

$$
\text{RL of staff point}=\text{RL of axis}+V-h
$$

$$
\begin{aligned}
\text{RL}_{A}&=1276.550+13.947-1.735=1288.762\ \text{m}\\
\text{RL}_{B}&=1276.550+23.798-2.000=1298.348\ \text{m}
\end{aligned}
$$

#### Step 3: Horizontal distance AB
The angle at P between the two lines is the difference of the bearings, $45^\circ 00'$ (30°30' and 75°30').

$$
\begin{aligned}
AB&=\sqrt{D_1^2+D_2^2-2D_1D_2\cos\gamma}\\
&=\sqrt{122.411^2+146.124^2-2\times122.411\times146.124\cos 45^\circ 00'}=105.073\ \text{m}
\end{aligned}
$$

#### Step 4: Gradient
$$
\text{Gradient}=\frac{\Delta H}{AB}=\frac{9.586}{105.073}=0.09123
$$

The line from A to B is rising: 1 in 11.0, i.e. 9.12 %, an angle of $5^\circ 12'46''$ with the horizontal.

**Answer:** AB = 105.07 m; RL of A = 1288.762 m, RL of B = 1298.348 m; difference in level = +9.586 m; gradient rising 1 in 11.0 (9.12 %, 5°13').

## Check yourself

The answer's key points as multiple-choice questions, with the correct option and why.

1. What is tacheometry?
   A. Determining horizontal distances and elevation differences from optical readings, without chaining or levelling
   B. Measuring areas with a planimeter
   C. Measuring only angles with a theodolite
   D. Levelling with a dumpy level and staff
   Answer: A. A tacheometer with stadia hairs and a staff gives distances and levels quickly. (Key point: Definition of tacheometry)

2. How is a staff point plotted from a tacheometric station?
   A. By its RL only
   B. By its bearing and horizontal distance at scale (polar), or by coordinates
   C. By its staff intercept directly at scale
   D. By chain offsets
   Answer: B. Each point is set off by protractor angle and scaled D from the station. (Key point: Plotting method)

3. For the anallatic lens in the problem, which formulae are used?
   A. D = Ks cos^2θ + C cosθ with C = 0.3
   B. D = Ks cosθ, V = Ks sinθ
   C. D = Ks cos^2θ, V = (1/2)Ks sin2θ with K = 100, C = 0
   D. D = Ks, V = 0
   Answer: C. The anallatic lens makes C = 0, and the staff is vertical. (Key point: Anallatic lens constants)

4. What is the RL of instrument axis at P?
   A. 1273.450 m
   B. 1275.000 m
   C. 1276.600 m
   D. 1276.550 m (RL 1275 + HI 1.55)
   Answer: D. RL of axis = RL of station + HI = 1275.000 + 1.550. (Key point: RL of instrument axis)

## Related in Surveying II

- [Anallatic tacheometer distance CD and gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-staff-cd-gradient-abcd) — Surveying II, chapter 2
- [Tacheometric RL of B, C, gradient BC](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-rl-b-c-gradient-bc) — Surveying II, chapter 2, asked 2×
- [Tacheometric RL of B, C, gradient BC](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-rl-b-c-gradient-bc-2) — Surveying II, chapter 2, asked 2×
- [Anallatic tacheometer A to B distance, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-bm-b-distance-gradient-970) — Surveying II, chapter 2
- [Anallatic tacheometer AB gradient from P](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-ab-gradient-percent-angle) — Surveying II, chapter 2, asked 3×
- [Anallatic tacheometer distance XY, gradient](https://www.bctengineer.com/question.md?key=semester_4%2Fsurveying_ii%3Aanallatic-xy-gradient-p) — Surveying II, chapter 2, asked 3×

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