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Area Computations Example

Medium DifficultyFE Surveying

Compute the area of a closed polygon (e.g. parcel or traverse) from the coordinates of its vertices.

Concept

For a closed polygon with vertices (X_{1}, Y_{1}), (X_{2}, Y_{2}), \dots, (X_{n}, Y_{n}) in order, the signed area can be computed with the coordinate (cross) formula: twice the area equals the sum of (X_{i + 1} - X_{i - 1}) \times Y_{i} over all vertices, with subscripts wrapping (e.g. X_{0} = X_{n}, X_{n + 1} = X_{1}). Take the absolute value for area. 2 A = i = 1 \sum n (X_{i + 1} - X_{i - 1}) (Y_{i}) Notation: X_{i}, Y_{i} = coordinates of vertex i, A = area. Positive result = clockwise order; negative = counterclockwise; area = ∣2 A ∣/2 .

A four-sided parcel has the following coordinates (ft) in order: P₁(100, 200), P₂(180, 250), P₃(140, 320), P₄(60, 270). Find: The area of the parcel in ft².

Vertices (ft): P_{1} (100, 200), P_{2} (180, 250), P_{3} (140, 320), P_{4} (60, 270)

2 A = i = 1 \sum n (X_{i + 1} - X_{i - 1}) (Y_{i}) (X_{2} - X_{4}) (Y_{1}) = (180 - 60) (200) = 24, 000 (X_{3} - X_{1}) (Y_{2}) = (140 - 100) (250) = 10, 000 (X_{4} - X_{2}) (Y_{3}) = (60 - 180) (320) = - 38, 400 (X_{1} - X_{3}) (Y_{4}) = (100 - 140) (270) = - 10, 800 2 A = 24, 000 + 10, 000 - 38, 400 - 10, 800 = - 15, 200 \Rightarrow A = 7600 ft^{2}

Area: A = 7600 ft^{2} (or about 0.174 acre if 43,560 ft² = 1 acre).

Area from coordinates: 2 A = \sum (X_{i + 1} - X_{i - 1}) Y_{i} (subscripts wrap)