Area Computations

Learning Objectives

  • Calculate area using the coordinate (shoelace) method.
  • Adjust area calculations to account for circular curves.
  • Apply the Double Meridian Distance (DMD) and Double Parallel Distance (DPD) methods.
  • Compute area of irregular boundaries using Trapezoidal and Simpson's 1/3 rules.

Accurate area determination is a primary goal of land surveying, essential for property valuation, deed descriptions, and construction earthworks. This lesson covers the analytical methods used to compute the area of polygons from field traverse data.

Area

The two-dimensional space occupied by a tract of land. In surveying, it is typically measured in square meters (m2m^2) or hectares (1 ha=10,000 m21 \text{ ha} = 10,000 \text{ m}^2).

Area by Coordinates (Shoelace Method)

Coordinate Method Principles

Given the coordinates of the vertices of a closed traverse (x1,y1),(x2,y2),…,(xn,yn)(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n), the area can be calculated using cross-multiplication. The vertices must be listed in consecutive order (clockwise or counter-clockwise).

Shoelace Formula (Expanded)

Calculates double the area by summing cross-products of sequential coordinates.

2A=∣(x1y2+x2y3+⋯+xny1)−(y1x2+y2x3+⋯+ynx1)∣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)|

Variables

SymbolDescriptionUnit
AATotal area of the polygonm2m^2
xxEasting (Departure) coordinate of a vertexm
yyNorthing (Latitude) coordinate of a vertexm

Shoelace Formula (Summation)

Compact representation of the coordinate area calculation.

A=12∣Σ(xiyi+1)−Σ(yixi+1)∣A = \frac{1}{2} | \Sigma (x_i y_{i+1}) - \Sigma (y_i x_{i+1}) |

Variables

SymbolDescriptionUnit
AATotal area of the polygonm2m^2
xix_iEasting of the i-th vertexm
yiy_iNorthing of the i-th vertexm

Interactive Coordinate Area

Interact with the simulation below to see how vertex coordinates determine the area.

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)
00224466881010ABCD
Drag vertices to modify the polygon

Area Involving Circular Curves

Adjusting Polygon Area for 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.

To find the true area, you must account for the circular segment.

Curve Adjustment Procedure

  1. Calculate the area of the polygon using the straight chord (usually via the coordinate method).
  2. Calculate the area of the circular segment formed by the arc and the chord using the radius RR and central angle Δ\Delta: Asegment=Asector−Atriangle=πR2Δ360∘−12R2sin⁡(Δ)A_{\text{segment}} = A_{\text{sector}} - A_{\text{triangle}} = \frac{\pi R^2 \Delta}{360^\circ} - \frac{1}{2} R^2 \sin(\Delta)
  3. Add the segment area if the curve bulges outward from the polygon (it adds land to the total area).
  4. Subtract the segment area if the curve bows inward into the polygon (it removes land from the total area).

Interactive Curve Adjustment

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

Area=12∣∑(XiYi+1−YiXi+1)∣\text{Area} = \frac{1}{2} |\sum(X_i Y_{i+1} - Y_i X_{i+1})|
Area=9000.0 sq units\text{Area} = 9000.0 \text{ sq units}
P3(100,100)

Interactive Area Calculator

Traverse Area Computation Tool

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

Traverse & Area Tool

Traverse Lines

1
2
3
4

Plot

Closure Error: 0.0000 m
Precision: 1:3,109,888,511,975,475
Area: 15000.00 m²
Start123End
* Y-axis is inverted for SVG rendering (North is up)

Area by Double Meridian Distance (DMD)

DMD Principles

The DMD method computes area based directly on the balanced latitudes and departures of the traverse, bypassing the need to compute arbitrary coordinates first.

DMD Calculation Rules

  1. DMD of the first course: Equal to the Departure of the first course.
  2. DMD of any other course: Equal to the DMD of the preceding course + Departure of the preceding course + Departure of the course itself.
  3. DMD of the last course: Equal to the negative of the Departure of the last course (used as a mathematical check).

Area by DMD

Calculates the total area using the summation of the product of DMDs and their corresponding latitudes.

2A=Σ(DMD×Latitude)2A = \Sigma (\text{DMD} \times \text{Latitude})A=12∣2A∣A = \frac{1}{2} |2A|

Variables

SymbolDescriptionUnit
AATotal area of the traversem2m^2
DMD\text{DMD}Double Meridian Distance for a specific coursem
Latitude\text{Latitude}Balanced latitude for a specific coursem

Double Parallel Distance (DPD)

DPD Principles

Similar to DMD but uses Latitudes instead of Departures.

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

Area with Irregular Boundaries

Approximating Irregular Boundaries

Used when one side of the area is an irregular curve (e.g., a river bank). Offsets (hh) are measured from a straight traverse line (baseline) to the boundary at regular intervals (dd).

1. Trapezoidal Rule

Linear Approximation

The Trapezoidal Rule assumes the boundary between adjacent offset points is a straight line.

Trapezoidal Rule

Calculates the area of irregular sections assuming straight line segments between offsets.

A=d(h1+hn2+h2+h3+⋯+hn−1)A = d \left(\frac{h_1 + h_n}{2} + h_2 + h_3 + \dots + h_{n-1}\right)

Variables

SymbolDescriptionUnit
AAArea between the traverse line and the irregular boundarym2m^2
ddCommon interval distance between offsetsm
h1,hnh_1, h_nEnd offsetsm
h2,h3,…h_2, h_3, \dotsIntermediate offsetsm

2. Simpson's 1/3 Rule

Parabolic Approximation

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

Simpson's 1/3 Rule

Calculates area assuming parabolic segments between offsets.

A=d3(h1+hn+4Σheven+2Σhodd)A = \frac{d}{3} \left(h_1 + h_n + 4\Sigma h_{\text{even}} + 2\Sigma h_{\text{odd}}\right)

Variables

SymbolDescriptionUnit
AAArea between the traverse line and the irregular boundarym2m^2
ddCommon interval distance between offsetsm
h1,hnh_1, h_nEnd offsetsm
Σheven\Sigma h_{\text{even}}Sum of even-numbered offsets (h_2, h_4, \dots)m
Σhodd\Sigma h_{\text{odd}}Sum of intermediate odd-numbered offsets (h_3, h_5, \dots)m

Planimeter Method

Mechanical Area Measurement

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

  • Principle: Based on Green's Theorem.
  • Use: Rapid determination of areas directly from maps, plans, or photographs (e.g., calculating the area of irregular parcels, lakes, or drainage basins).
Key Takeaways
  • Coordinate Method (Shoelace): The general and most robust method for determining the area of any closed polygon defined by coordinates.
  • Curve Adjustments: Add or subtract the area of the circular segment (Asector−AtriangleA_{\text{sector}} - A_{\text{triangle}}) depending on the boundary's bulge direction relative to the straight chord.
  • DMD/DPD: Classic methods to calculate area directly from balanced Latitudes and Departures.
  • Trapezoidal Rule: Provides a straight-line approximation for measuring area with irregular boundaries.
  • Simpson's Rule: Uses a parabolic approximation for irregular boundaries; it is more accurate than the Trapezoidal rule but strictly requires an odd number of offsets.
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