--- title: "Tangential method distance and elevation" subject: "Surveying II" chapter: 2 key: "semester_4/surveying_ii:tangential-method-distance-measurement" source: https://www.bctengineer.com/q/semester_4/surveying_ii/tangential-method-distance-measurement site: BCT Engineering --- # Tangential method distance and elevation Surveying II · Chapter 2: Tacheometry - Asked in: 2078 Baisakh, 4 marks - Video: [Tangential Method of Tacheometric Surveying](https://www.youtube.com/watch?v=Vt0gh_NqghA) — CIVIL Solved - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_4/surveying_ii/tangential-method-distance-measurement - Whole chapter: https://www.bctengineer.com/md/semester_4/surveying_ii/chapter-2 ## Question Discuss measurement of horizontal and vertical distance by tangential method. ## Short answer Measure vertical angles θ1 and θ2 to two targets a known distance s apart on a staff, without using stadia hairs. Horizontal distance D = s/(tanθ1 − tanθ2) and vertical component V = D tanθ2. The staff station RL equals RL of axis + V − h, where h is the lower target height above ground. ## Answer In the **tangential method** the stadia hairs are not used. The telescope is pointed at two targets (vanes) on a staff, a known distance $s$ apart, and the two vertical angles are measured. The distance is calculated from the tangents of these angles. It is used when the telescope has no stadia hairs, or when the distance is too large for the stadia intercept to be read accurately. ```text _ B (upper target) _ - | theta1 - | s O ---------- A (lower target) theta2 \_ | axis ----------\__ | horizontal D ``` Let $D$ be the horizontal distance of the staff from the instrument axis, and $V$ the vertical height of the lower target above (or below) the axis. #### Horizontal distance and vertical distance For two angles of elevation, $\theta_1$ (upper target) and $\theta_2$ (lower target), with the vanes $s$ apart: $$ D=\frac{s}{\tan\theta_1-\tan\theta_2},\qquad V=D\tan\theta_2 $$ $$ \text{RL of staff station}=\text{RL of axis}+V-h $$ where $h$ is the height of the lower vane above the ground at the staff station. For two angles of depression, $\alpha_1$ (lower target) $>\alpha_2$ (upper target): $$ D=\frac{s}{\tan\alpha_1-\tan\alpha_2},\qquad \text{RL}=\text{RL of axis}-D\tan\alpha_2-s-h $$ For one angle of elevation $\theta$ (upper target) and one of depression $\alpha$ (lower target): $D=s/(\tan\theta+\tan\alpha)$ and RL $=$ RL of axis $-D\tan\alpha-h$. The method needs only the vertical circle of the theodolite and works even at long distances, but the angles must be measured carefully, because $\tan\theta_1-\tan\theta_2$ is small and a small error in either angle causes a large error in $D$. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. What does the tangential method use instead of stadia hairs? A. Two targets on a staff a known distance apart and vertical angles to them B. A subtense bar C. A chain and tape D. A level and staff Answer: A. Two vertical angles to targets s apart give distance through tangents. (Key point: Principle) 2. For two angles of elevation theta1 (upper) and theta2 (lower), D equals A. s/(tan theta1 + tan theta2) B. s (tan theta1 - tan theta2) C. s/(tan theta1 - tan theta2) D. s/(tan theta1 x tan theta2) Answer: C. D = s/(tan theta1 - tan theta2); V = D tan theta2. (Key point: Distance formula) 3. What is the RL of the staff station for angles of elevation? A. RL axis - V - h B. RL axis + V - h C. RL axis + V + h D. RL axis + D - h Answer: B. RL = RL of axis + V - h, with h the height of the lower vane above the station. (Key point: RL for elevation) 4. Why does the method need careful angle measurement? A. Error in D depends on d(theta)/(tan theta1 - tan theta2) B. The angles are measured in radians C. Hair interval is unknown D. The staff is horizontal Answer: A. Because the denominator is a small difference of tangents, small angle errors affect D greatly. 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