---
title: "Area by coordinates method"
subject: "Surveying I"
chapter: 9
key: "semester_3/surveying_i:area-by-coordinates-method"
source: https://www.bctengineer.com/q/semester_3/surveying_i/area-by-coordinates-method
site: BCT Engineering
---

# Area by coordinates method

Surveying I · Chapter 9: Computation of Area and Volume

- Asked in: 2073 Shrawan, 4 marks
- Video: [Area by Coordinates - Example](https://www.youtube.com/watch?v=0AG-k3DXfAg) — Fech
- Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14
- Page: https://www.bctengineer.com/q/semester_3/surveying_i/area-by-coordinates-method
- Whole chapter: https://www.bctengineer.com/md/semester_3/surveying_i/chapter-9

## Question

Work out the co-ordinates method for finding area.

## Short answer

List the (x, y) coordinates of polygon vertices in order, repeating the first at the end. Area = 1/2 | Σ (x_i y_{i+1} - x_{i+1} y_i) |. This gives the exact area enclosed by straight-line boundaries.

## Answer

When the coordinates (northing $N$ and easting $E$, or $x$ and $y$) of the stations of a closed traverse or boundary are known, the area is found directly from them.

#### Formula
Number the stations $1, 2, \dots, n$ in order round the figure (clockwise or anticlockwise), with the coordinates $(x_i, y_i)$ and $(x_{n+1}, y_{n+1}) = (x_1, y_1)$.

$$
\text{Area} = \frac{1}{2}\left|\sum_{i=1}^{n} \left(x_i\,y_{i+1} - x_{i+1}\,y_i\right)\right|
$$

or, in the form with the coordinate differences,

$$
\text{Area} = \frac{1}{2}\left|\sum x_i\,(y_{i+1} - y_{i-1})\right|
$$

#### Procedure
1. Compute the adjusted latitudes and departures, and the coordinates of every station from a convenient origin (preferably so that all coordinates are positive).
2. Write the coordinates in order in two columns and repeat the first station at the end.
3. Multiply each $x$ by the next $y$ (downward products) and each $y$ by the next $x$ (upward products).
4. Take the difference of the two sums and halve it. The sign depends on the direction of travel; the numerical value is the area.

#### Example
Stations: A (0, 0), B (80, 30), C (100, 100), D (20, 90) in (E, N).

| Pair | $x_i y_{i+1}$ | $x_{i+1} y_i$ |
|---|---|---|
| A to B | 0 × 30 = 0 | 80 × 0 = 0 |
| B to C | 80 × 100 = 8000 | 100 × 30 = 3000 |
| C to D | 100 × 90 = 9000 | 20 × 100 = 2000 |
| D to A | 20 × 0 = 0 | 0 × 90 = 0 |
| **Sum** | **17000** | **5000** |

$$
\text{Area} = \frac{1}{2}(17000 - 5000) = 6000\ \text{m}^2
$$

The method is exact for a polygon and is suitable for a computer or calculator, with no need for dividing the figure into parts.

## Check yourself

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

1. Which formula gives area from coordinates?
   A. sum (x_i y_i)
   B. 1/2 sum (x_i + y_i)
   C. sum x_i x y_i / n
   D. 1/2 |sum (x_i y_(i+1) - x_(i+1) y_i)|
   Answer: D. The cross-product sum of consecutive stations, halved, gives the area. (Key point: Formula)

2. Why is the first station repeated at the end of the list?
   A. To close the figure so the last pair is included
   B. To double the area
   C. To check bearings
   D. To fix the origin
   Answer: A. The product for the last-to-first pair must be included. (Key point: Closing the polygon)

3. What does the sign of the result depend on?
   A. The scale
   B. The direction of travel; the numerical value is the area
   C. The origin only
   D. The number of stations
   Answer: B. Clockwise or anticlockwise gives opposite signs, so the absolute value is taken. (Key point: Sign and value)

4. For A(0,0), B(80,30), C(100,100), D(20,90), what is the area?
   A. 12000 m2
   B. 3000 m2
   C. 6000 m2
   D. 8000 m2
   Answer: C. (17000 - 5000)/2 = 6000 m2. (Key point: Worked example)

## Related in Surveying I

- [Area by coordinates and DMD (AB -370)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-coordinate-and-dmd-minus-370) — Surveying I, chapter 9, asked 4×
- [Area by total coordinates and DMD (AB 90.29)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Admd-and-coordinates-ab-90-29) — Surveying I, chapter 9
- [Area of traverse by DMD and coordinates (N/S/E/W)](https://www.bctengineer.com/question.md?key=semester_3%2Fsurveying_i%3Aarea-closed-traverse-nswe-108) — Surveying I, chapter 9

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