Area by Coordinates (Shoelace)

A = \frac{1}{2} i = 1 \sum n (x_{i} y_{i + 1} - x_{i + 1} y_{i})

Traverse the vertices in order (clockwise or counterclockwise) and close back to the first point (x_{n+1}=x_1). The surveyor’s (DMD or coordinate) formula is the shoelace identity. Units: if x,y are feet, A is ft² (÷ 43 560 for acres).

$A$: Enclosed area
$x_{i}, y_{i}$: Easting and northing of vertex i
$n$: Number of sides

Step-by-step solved example

Lot vertices (ft): (0,0), (50,0), (50,20), (0,20). Find area.

Surveyor's formula: twice area from coordinate cross products.

1. Shoelace terms
Σ(x_i y_{i+1} − x_{i+1} y_i) = (0·0−50·0)+(50·20−50·0)+(50·20−0·20)+(0·0−0·20) = 0+1000+1000+0 = 2000.
2. Area
A = ½|2000| = 1000 ft² (50×20 rectangle).
$A = 1000 ft^{2}$

Answer: A = 1000 ft²

10 practice questions

0/10 correct

1.Triangle (0,0), (40,0), (0,30) ft. A =

2.You must list the first point again at the end in order to

3.43 560 ft² equals

4.Reversing vertex order (CW vs CCW) makes A

5.A 2.50 acre parcel is

6.Coordinate area assumes the sides are

7.DMD (double meridian distance) method is equivalent to

8.Vertices (m): (2,1), (5,1), (5,5), (2,5). A =

9.If all y_i = 0 the shoelace area is

10.1 hectare = 10 000 m² is about