Area Computations

Area Computations

Determining the area of a tract of land is one of the primary objectives of land surveying.

Area by Coordinates

Given the coordinates of the vertices of a closed traverse ($x_1, y_1$), ($x_2, y_2$), ..., ($x_n, y_n$).

Formula (Shoelace Formula)

$2A = |(x_1 y_2 + x_2 y_3 + \dots + x_n y_1) - (y_1 x_2 + y_2 x_3 + \dots + y_n x_1)|$

Where:

• $x$: Eastings (Departures)
• $y$: Northings (Latitudes)

The vertices must be listed in consecutive order (clockwise or counter-clockwise).

Area by Double Meridian Distance (DMD)

This method is based on the balanced latitudes and departures of the traverse.

1. DMD Rules:

   • DMD of the first course = Departure of the first course.
   • DMD of any other course = DMD of the preceding course + Departure of the preceding course + Departure of the course itself.
   • DMD of the last course = -Departure of the last course.

2. Double Area ($2A$): $2A = \Sigma (\text{DMD} \times \text{Latitude})$

3. Area ($A$): $A = \frac{1}{2} |2A|$

Area with Irregular Boundaries

Used when one side of the area is an irregular curve (e.g., a river bank). Offsets ($h$) are measured from a traverse line at regular intervals ($d$).

1. Trapezoidal Rule

Assumes the boundary between offsets is a straight line.

$A = d \left(\frac{h_1 + h_n}{2} + h_2 + h_3 + \dots + h_{n-1}\right)$

Where:

• $d$: Common interval distance.
• $h_1, h_n$: End offsets.
• $n$: Number of offsets.

2. Simpson's 1/3 Rule

Assumes the boundary between offsets is a parabolic arc. More accurate than Trapezoidal Rule but requires an odd number of offsets (even number of intervals).

$A = \frac{d}{3} \left(h_1 + h_n + 4\Sigma h_{even} + 2\Sigma h_{odd}\right)$

Where:

• $h_{even}$: Sum of even offsets ($h_2, h_4, \dots$).
• $h_{odd}$: Sum of odd offsets ($h_3, h_5, \dots$).

Solved Problems