Area Computations - Theory & Concepts

Area Computations

The two-dimensional space occupied by a tract of land. In surveying, it is typically measured in square meters ( $m^2$ ) or hectares ( $1 \text{ ha} = 10,000 \text{ m}^2$ ).

Area by Coordinates (Shoelace Method)

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

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)|

A = \frac{1}{2} | \Sigma (x_i y_{i+1}) - \Sigma (y_i x_{i+1}) |

Where:

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

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

Shoelace Method Simulator

Drag the points to change the shape of the polygon. The area is automatically calculated using the Coordinate (Shoelace) Method.

Total Area

0.0 sq units

Coordinates:

Point A(2, 2)

Point B(8, 2)

Point C(6, 6)

Point D(4, 5)

Drag vertices to modify the polygon

Area Involving Circular Curves

When a property boundary includes a circular curve (e.g., along a curved road), the standard coordinate method only gives the area of the straight-line polygon defined by the chord connecting the curve's endpoints.

Adjusting for Curves

Calculate the area of the polygon using the straight chord (usually via the coordinate method).
Calculate the area of the circular segment formed by the arc and the chord using the radius $R$ and central angle $\Delta$ :
$A_{segment} = A_{sector} - A_{triangle} = \frac{\pi R^2 \Delta}{360^\circ} - \frac{1}{2} R^2 \sin(\Delta)$
Add the segment area if the curve bulges outward from the polygon (it adds land to the total area).
Subtract the segment area if the curve bows inward into the polygon (it removes land from the total area).

Move the interactive point on the polygon to see how it affects the total area calculated using the coordinates method.

Area by Coordinates Simulator

Visualize how moving a polygon vertex affects its calculated area.

Point 3 X100.0

Point 3 Y100.0

Results

\text{Area} = \frac{1}{2} |\sum(X_i Y_{i+1} - Y_i X_{i+1})|

\text{Area} = 9000.0 \text{ sq units}

Interactive Area Calculator

Use the tool below to input a traverse (lengths and directions) and calculate its area.

Traverse & Area Tool

Traverse Lines

Length (m)

Azimuth (°)

Length (m)

Azimuth (°)

Length (m)

Azimuth (°)

Length (m)

Azimuth (°)

Plot

Closure Error: 0.0000 m

Precision: 1:3,109,888,511,975,475

Area: 15000.00 m²

* Y-axis is inverted for SVG rendering (North is up)

Area by Double Meridian Distance (DMD)

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

DMD Rules

DMD of the first course: Equal to the Departure of the first course.
DMD of any other course: Equal to the DMD of the preceding course + Departure of the preceding course + Departure of the course itself.
DMD of the last course: Equal to the negative of the Departure of the last course (Check).

Double Area ( $2A$ ):

2A = \Sigma (\text{DMD} \times \text{Latitude})

Area ( $A$ ):

A = \frac{1}{2} |2A|

Double Parallel Distance (DPD)

Similar to DMD but uses Latitudes instead of Departures.

DPD of first course: Equal to the Latitude of the first course.
2A: $\Sigma (\text{DPD} \times \text{Departure})$ .

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.

Trapezoidal Rule

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).

Simpson's Rule

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 ( $h_3, h_5, \dots$ ).

Planimeter Method

A Planimeter is a mechanical or digital instrument used to measure the area of a plane figure by tracing its perimeter.

Principle: Based on Green's Theorem.
Use: Rapid determination of areas from maps or plans (e.g., irregular parcels, lakes).

Key Takeaways

Coordinate Method: General method for any polygon.
DMD/DPD: Based on balanced Latitudes and Departures.
Trapezoidal Rule: Straight-line approximation for irregular boundaries.
Simpson's Rule: Parabolic approximation; requires odd number of offsets. More accurate for curved boundaries.
Curve Adjustments: Add or subtract the area of the circular segment ( $A_{sector} - A_{triangle}$ ) depending on the boundary's bulge direction.

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