--- title: "Offsets area: trapezoidal, Simpson, coordinate" subject: "Surveying I" chapter: 9 key: "semester_3/surveying_i:area-offsets-curved-boundary-trapezoidal-simpson-coordinate" source: https://www.bctengineer.com/q/semester_3/surveying_i/area-offsets-curved-boundary-trapezoidal-simpson-coordinate site: BCT Engineering --- # Offsets area: trapezoidal, Simpson, coordinate Surveying I · Chapter 9: Computation of Area and Volume - Asked in: 2081 Bhadra, 1+1+2 marks - Video: [FE Review - Surveying - Area Computations Using Trapezoidal Rule and Simpson's](https://www.youtube.com/watch?v=z4iNhP9_rhQ) — FE MADE EASY - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/surveying_i/area-offsets-curved-boundary-trapezoidal-simpson-coordinate - Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9 ## Question A series of offsets were taken from a chain line to a curved boundary line at intervals of 5 m in the following order: 0, 1.68, 5.70, 7.60, 6.12, 10.84, 4.75, 0.90. Compute the area between the chain line, the curved boundary line and the end offset by trapezoidal method, Simpson's rule and coordinate method. ## Short answer Trapezoidal rule gives 185.70 m². Simpson's 1/3 rule applied to the first six intervals and trapezoidal to the last gives 195.58 m². The coordinate method also yields 185.70 m². ## Answer **Given:** 8 offsets at a common interval $d = 5$ m, so the length of the chain line = 7 × 5 = 35 m. #### (i) Trapezoidal rule $$ \text{Area} = d\left[\frac{o_0 + o_n}{2} + o_1 + o_2 + \dots + o_{n-1}\right] $$ Sum of the intermediate offsets = 36.69 m; (first + last)/2 = (0.00 + 0.90)/2 = 0.450 m. $$ \text{Area} = 5\times\left[0.450 + 36.69\right] = 185.700\ \text{m}^2 $$ #### (ii) Simpson's one-third rule There are 8 offsets, i.e. 7 (an odd number of) intervals, so Simpson's rule cannot be used for the whole length. It is applied to the first 6 intervals (offsets 0 to 6), and the last strip is found by the trapezoidal rule. $$ \text{Area}_1 = \frac{5}{3}\left[(0.00 + 4.75) + 4(20.12) + 2(11.82)\right] = 181.450\ \text{m}^2 $$ $$ \text{Area}_2 = 5\times\frac{4.75 + 0.90}{2} = 14.125\ \text{m}^2 $$ $$ \text{Total area} = 181.450 + 14.125 = 195.575\ \text{m}^2 $$ #### (iii) Coordinate method Take the chain line as the x-axis with the first offset at chainage 0. The corner points of the area, taken in order round the boundary, are: | Point | $x$ (m) | $y$ (m) | |---|---|---| | Chain start | 0.00 | 0.00 | | Chain end | 35.00 | 0.00 | | offset 8 | 35.00 | 0.90 | | offset 7 | 30.00 | 4.75 | | offset 6 | 25.00 | 10.84 | | offset 5 | 20.00 | 6.12 | | offset 4 | 15.00 | 7.60 | | offset 3 | 10.00 | 5.70 | | offset 2 | 5.00 | 1.68 | | offset 1 | 0.00 | 0.00 | $$ \text{Area} = \frac{1}{2}\left|\sum \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right| $$ Evaluating the sum with these points gives Area = **185.700 m²**. (The coordinate method, with straight lines between the offsets, gives the same figure as the trapezoidal rule, since both treat the boundary as a series of straight segments.) **Answer:** trapezoidal = **185.70 m²**, Simpson's = **195.58 m²**, coordinate method = **185.70 m²**. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. With 8 offsets at 5 m interval, what is the length of the chain line? A. 40 m B. 30 m C. 45 m D. 35 m Answer: D. There are 7 intervals of 5 m, so 35 m. (Key point: Chain length) 2. Why can Simpson's rule not be applied to the whole length here? A. There are 7 intervals, an odd number B. The offsets are unequal C. The first offset is zero D. The interval is 5 m Answer: A. Simpson's rule needs an even number of intervals; the last strip uses the trapezoidal rule. (Key point: Simpson limitation) 3. What is the trapezoidal area? A. 195.57 m2 B. 185.70 m2 C. 181.45 m2 D. 200.00 m2 Answer: B. 5 x [0.45 + 36.69] = 185.70 m2. (Key point: Trapezoidal result) 4. Why does the coordinate method give the same value as the trapezoidal rule? A. Both use parabolas B. Both ignore the end offsets C. Both treat the boundary as straight segments between offsets D. It is a coincidence Answer: C. Straight lines between offset points form the same trapezia. (Key point: Coordinate method) ## Related in Surveying I - [Area by coordinate method from offsets](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-coordinate-method-chainage-3-75) — Surveying I, chapter 9 - [Area between boundary, offsets and base line](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-chainage-0-12-20-25-34-42-52) — Surveying I, chapter 9 - [Area enclosed by chain line and hedge](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-hedge-5m-interval-simpson-trapezoidal) — Surveying I, chapter 9 - [Area from offsets, 10 m interval](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-4-6-7-2-9-6-6-4) — Surveying I, chapter 9 - [Area from eight offsets at 20 m interval](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-20m-interval-eight-offsets) — Surveying I, chapter 9, asked 2× - [Area from twelve offsets at 20 m interval](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-offsets-twelve-offsets-4-19) — Surveying I, chapter 9, asked 2× ## Related in other subjects - [Line spectra of 12 + 6sin(140πt+30°) - 9cos(80πt-70°)](https://www.bctengineer.com/question.md?key=semester_5%2Fdata_communication%3Aline-spectrum-12-6sin) — Data Communication, chapter 3 - [Stability: NS 320/70, F 155/68](https://www.bctengineer.com/question.md?key=semester_4%2Fengineering_geology_ii%3Astability-ns320-70-f155-68) — Engineering Geology II, chapter 6 - [Cable size MDB to SDB1 (70 kW) and SDB2 (90 kW)](https://www.bctengineer.com/question.md?key=semester_6%2Findustrial_power_distribution_and_illumination%3Acable-mdb-sdb-70kw-90kw) — Industrial Power Distribution and Illumination, chapter 5, asked 5× - [Arch 100 m rise 12 m: rolling UDL 60 m, section 10 m](https://www.bctengineer.com/question.md?key=semester_4%2Ftheory_of_structures_i%3Aarch-100m-rise12-udl-60m-10m) — Theory of Structures I, chapter 6