--- title: "Distance measurement on sloping ground" subject: "Surveying I" chapter: 2 key: "semester_3/surveying_i:distance-on-sloping-ground" source: https://www.bctengineer.com/q/semester_3/surveying_i/distance-on-sloping-ground site: BCT Engineering --- # Distance measurement on sloping ground Surveying I · Chapter 2: Distance Measurements - Asked in: 2072 Chaitra, 4 marks - Video: [Methods of Chaining on Sloping Ground by Direct and Indirect Method | Stepping and Clinometer method](https://www.youtube.com/watch?v=rMrJ7XD0bx4) — Mukesh Kumar - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/distance-on-sloping-ground - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-2 ## Question Explain distance measurement in sloping ground. ## Short answer Horizontal distance on sloping ground is obtained by stepping (holding the tape horizontal with a plumb bob) or by measuring slope length L and reducing it: D = √(L² − h²) or D = L cosθ. Hypotenuse allowance adds an extra length so that each chain length equals one horizontal chain. ## Answer Horizontal distance is required for plotting, but on sloping ground the tape follows the slope, so the measurement must be reduced to the horizontal. There are two ways: direct (by stepping) and indirect (measuring along the slope and applying a correction). #### 1. Direct method: stepping The line is measured in short horizontal steps, as shown. The tape is held horizontal, with one end resting on the ground and the other end transferred to the ground by a plumb bob. The line is measured downhill when possible, so that the plumbed end is held at a convenient height. The steps are added to get the total horizontal distance. ```text A o------- step 1 (tape horizontal) | plumb bob o------- step 2 | o------- step 3 -> B ``` Limitations: it is slow, plumbing errors build up, and it is suited only to short steps on steep ground. #### 2. Indirect method: measuring along the slope and calculating The slope length $L$ is measured and the horizontal length $D$ is found by one of the following: - Height difference $h$ from levelling: $D = \sqrt{L^{2} - h^{2}}$, or the slope correction $C_h = -\dfrac{h^{2}}{2L}$ (approx.). - Slope angle $\theta$ measured with a clinometer or theodolite: $D = L\cos\theta$, correction $C_h = -L(1-\cos\theta)$. - Hypotenuse allowance: used in chain surveying on slopes. At the end of each chain length laid along the slope, the arrow is set forward by an extra length (the allowance), so that each chain length then corresponds to exactly one chain length (30 m) horizontally. For a slope $\theta$ and a 30 m chain, the allowance is $30\sec\theta - 30 = 30(\sec\theta - 1)$. For example, at $\theta = 10^\circ$ the allowance is 0.46 m. #### Example Slope length $L = 30$ m and $h = 2$ m. Then $D = \sqrt{900 - 4} = 29.933$ m (correction $-0.067$ m). The approximate formula gives $-4/60 = -0.067$ m. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In direct stepping on sloping ground, the tape is held A. Along the slope B. Horizontal, with one end plumbed C. Vertical D. On the ground only Answer: B. Short horizontal steps are measured with a plumb bob and added up. (Key point: Stepping) 2. With slope length L and height difference h, the horizontal distance D is A. sqrt(L^2 + h^2) B. sqrt(L^2 - h^2) C. L + h D. L / h Answer: B. L is the hypotenuse, so D = sqrt(L^2 - h^2). (Key point: Slope reduction formula) 3. For a 30 m chain on slope theta, the hypotenuse allowance is A. 30 (sec theta - 1) B. 30 cos theta C. 30 sin theta D. 30 (1 - cos theta) Answer: A. The chain must be 30 sec theta long on the slope to span 30 m horizontally. (Key point: Hypotenuse allowance) 4. L = 30 m and h = 2 m: what is the slope correction? A. +0.067 m B. -0.067 m C. -0.67 m D. 0 Answer: B. -h^2/2L = -4/60 = -0.067 m, giving D = 29.933 m. 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