--- title: "Area by coordinate method from offsets" subject: "Surveying I" chapter: 9 key: "semester_3/surveying_i:area-coordinate-method-chainage-3-75" source: https://www.bctengineer.com/q/semester_3/surveying_i/area-coordinate-method-chainage-3-75 site: BCT Engineering --- # Area by coordinate method from offsets Surveying I · Chapter 9: Computation of Area and Volume - Asked in: 2069 Chaitra, 8 marks - Video: [Area by coordinates : Irregular interval offsets : Surveying](https://www.youtube.com/watch?v=PGOsDJbIfoc) — The Classroom - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/area-coordinate-method-chainage-3-75 - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9 ## Question Calculate the area by the coordinate method from the following perpendicular offsets taken from a chain line to a boundary. | Chainage (m) | 0.00 | 3.75 | 6.50 | 11.30 | 16.45 | |---|---|---|---|---|---| | Offset (m) | 1.45 | 2.50 | 2.95 | 2.10 | 2.35 | ## Short answer Take the chain line as x-axis and form a closed polygon with the chain ends and offset endpoints. Coordinate area = ½|Σ(x_i y_{i+1} − x_{i+1} y_i)| = 38.479 m². ## Answer **Method.** The chain line is taken as the x-axis and the first offset position as the origin. The ends of the offsets and the two chain-line corners form a closed polygon whose area is found from its coordinates: $$ \text{Area} = \frac{1}{2}\left|\sum \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right| $$ #### Coordinates (taken in order round the area) | Point | $x$ (m) | $y$ (m) | |---|---|---| | Chain start | 0.00 | 0.00 | | Chain end | 16.45 | 0.00 | | offset 5 | 16.45 | 2.35 | | offset 4 | 11.30 | 2.10 | | offset 3 | 6.50 | 2.95 | | offset 2 | 3.75 | 2.50 | | offset 1 | 0.00 | 1.45 | #### Calculation | Pair | $x_i y_{i+1}$ | $x_{i+1} y_i$ | Difference | |---|---|---|---| | P1→P2 | 0.0000 | 0.0000 | 0.0000 | | P2→P3 | 38.6575 | 0.0000 | 38.6575 | | P3→P4 | 34.5450 | 26.5550 | 7.9900 | | P4→P5 | 33.3350 | 13.6500 | 19.6850 | | P5→P6 | 16.2500 | 11.0625 | 5.1875 | | P6→P7 | 5.4375 | 0.0000 | 5.4375 | | P7→P1 | 0.0000 | 0.0000 | 0.0000 | | **Sum** | | | **76.9575** | $$ \text{Area} = \frac{1}{2}\times |76.9575| = 38.4788\ \text{m}^2 $$ **Check (trapezoidal strips):** the four strips give 7.4063 + 7.4938 + 12.1200 + 11.4588 = 38.4788 m². **Answer:** area = **38.479 m²**. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. How is the coordinate system set up for the offsets? A. Offsets as x-axis and chain line as y-axis B. Chain line as x-axis, offsets as y-values, first offset position as origin C. North-south as x-axis using a compass D. Largest offset as origin Answer: B. Chainage is x along the chain line and offset length is y. (Key point: Coordinate axes) 2. What closed figure is used to compute the area? A. Chain-line ends plus the offset ends taken in order round the area B. Only the offset ends joined together C. A triangle from the first, middle and last offsets D. A rectangle of length times maximum offset Answer: A. The two chain-line corners and the offset ends form a closed polygon; the coordinate formula needs a closed boundary taken in order. (Key point: Closed polygon) 3. Which formula is used for the area? A. sum (x_i y_i) B. 1/2 sum (x_i + y_i) C. d/3 sum of offsets D. 1/2 |sum (x_i y_(i+1) - x_(i+1) y_i)| Answer: D. The cross-product (shoelace) sum is halved and its absolute value taken. (Key point: Coordinate formula) 4. How does the answer verify the result? A. By repeating with Simpson's rule only B. By comparing with the perimeter C. By adding four trapezoidal strips, giving the same 38.479 m^2 D. By no check Answer: C. The trapezoidal strips between successive offsets agree with the coordinate area of about 38.48 m^2. 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