8.3 Area Calculations (Triangles, Trapezoids, Rectangles, and Simple Polygons)

8.3 Area Calculations (Triangles, Trapezoids, Rectangles, and Simple Polygons)

Area calculation is one of the primary duties of survey technicians, essential for property boundary descriptions, land valuation, deed verification, tax mapping, and earthwork volume estimating. Whether evaluating simple geometric lot shapes or computing the area of complex closed boundary polygons, surveyors must perform calculations accurately in both Square Feet ($ ext{sq ft}$ or $ ext{ft}^2$) and Acres ($ ext{ac}$). This section details the fundamental formulas for standard geometric shapes, introduces the coordinate area (shoelace) method and Double Meridian Distance (DMD) technique for closed traverses, and highlights unit conversion rules for the CST Level I examination.

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1. Units of Area and Land Measurement Conversions

In U.S. Customary survey practice, land areas are measured in square feet or acres.

The Standard Acre Conversion Factor

$1\text{ Acre} = 43,560\text{ Square Feet}$

Historical Origin of the Acre:

The unit originates from historical Gunter's chain surveying, where 1 chain = 66 feet and 1 furlong = 660 feet. An acre was defined as 1 square furlong, or 10 square chains: $1\text{ Acre} = 10\text{ chains}^2 = 10 \times (66\text{ ft} \times 66\text{ ft}) = 10 \times 4,356\text{ sq ft} = 43,560\text{ sq ft}$

Converting Square Feet to Acres:

$\text{Area in Acres} = \frac{\text{Area in Square Feet}}{43,560}$

Example: A lot with an area of $125,480\text{ sq ft}$ converted to acres: $\text{Acres} = \frac{125,480}{43,560} = 2.88062... \approx 2.881\text{ Acres}$

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2. Standard Geometric Formulas

A. Rectangles and Squares

For a rectangular boundary with perpendicular sides of length ($L$) and width ($W$): $\text{Area} = \text{Length} \times \text{Width} = L \times W$

+------------------------+
|                        | Width (W)
+------------------------+
        Length (L)

B. Triangles

Surveyors encounter triangular parcels or subdivide complex polygons into simpler triangles using three key formulas based on available field data:

        /|
       / |
    a /  | h (Height)
     /   |
    /____|____
       b (Base)

1. Base-Height Formula (When perpendicular height $h$ and base $b$ are known): $\text{Area} = \frac{1}{2} \times b \times h$

2. Side-Angle-Side (SAS) Formula (When two sides $a$ and $b$ and their included angle $C$ are known): $\text{Area} = \frac{1}{2} \times a \times b \times \sin(C)$

3. Heron's Formula (When all three side lengths $a$, $b$, and $c$ are known): $s = \frac{a + b + c}{2} \quad (\text{Semi-perimeter})$ $\text{Area} = \sqrt{s(s - a)(s - b)(s - c)}$

Step-by-Step Triangle Example (SAS Formula):

Problem: Calculate the area of a triangular lot with side $a = 185.00\text{ ft}$, side $b = 240.00\text{ ft}$, and included angle $C = 52° 15' 00"$.

• Step 1: Convert angle to decimal degrees: $52° 15' = 52.25°$.
• Step 2: Calculate $\sin(52.25°) = 0.7906896$.
• Step 3: Apply SAS formula: $\text{Area} = \frac{1}{2} \times 185.00 \times 240.00 \times 0.7906896 = 22,200 \times 0.7906896 = 17,553.31\text{ sq ft}$
• Step 4: Convert to acres: $\text{Acres} = \frac{17,553.31}{43,560} = 0.403\text{ Acres}$

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C. Trapezoids

Trapezoids occur frequently in road cross-sections, right-of-way strips, and boundary lots bounded by two parallel lines ($b_1$ and $b_2$) separated by perpendicular distance ($h$).

$\text{Area} = \frac{b_1 + b_2}{2} \times h = \frac{1}{2} (b_1 + b_2) h$

       b1 (Top Base)
      +----------+
     / |          \
    /  | h         \
   +---|------------+
     b2 (Bottom Base)

Trapezoid Step-by-Step Example:

Problem: A drainage easement forms a trapezoid with parallel sides measuring $75.00\text{ ft}$ and $115.00\text{ ft}$, and a perpendicular distance between them of $40.00\text{ ft}$. Compute the area.

• Step 1: Average the two bases: $\frac{75.00 + 115.00}{2} = \frac{190.00}{2} = 95.00\text{ ft}$.
• Step 2: Multiply by height: $\text{Area} = 95.00 \times 40.00 = 3,800.00\text{ sq ft}$.

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3. Coordinate Area Method (Shoelace Formula)

When a closed polygon's boundary points are defined by rectangular coordinates $(N_i, E_i)$, the exact enclosed area is computed using the Coordinate Method (also known as the Shoelace Formula or Gauss's Area Formula).

Point 1 (N1, E1) ----> Point 2 (N2, E2)
       ^                      |
       |                      v
Point 4 (N4, E4) <---- Point 3 (N3, E3)

Tabular Execution Steps:

1. Arrange station coordinates in sequential order around the perimeter (either clockwise or counter-clockwise).
2. List the starting point coordinates again at the bottom of the table to close the loop.
3. Cross-multiply down-right (Northings $\times$ next Eastings) and sum the products ($\sum \text{Product}_{NE}$).
4. Cross-multiply down-left (Eastings $\times$ next Northings) and sum the products ($\sum \text{Product}_{EN}$).
5. Subtract the total left sum from the total right sum, take the absolute value, and divide by 2.

$\text{Double Area} = \left| \sum (N_i \times E_{i+1}) - \sum (E_i \times N_{i+1}) \right|$ $\text{Area} = \frac{\text{Double Area}}{2}$

Worked Coordinate Area Example:

Problem: Calculate the area of a 4-point closed survey boundary with coordinates:

• Station A: $N = 100.00\text{ ft}, E = 100.00\text{ ft}$
• Station B: $N = 100.00\text{ ft}, E = 400.00\text{ ft}$
• Station C: $N = 300.00\text{ ft}, E = 500.00\text{ ft}$
• Station D: $N = 400.00\text{ ft}, E = 100.00\text{ ft}$

• Step 1: Calculate difference: $|160,000 - 340,000| = |-180,000| = 180,000\text{ sq ft}$ (Double Area).
• Step 2: Divide by 2: $\text{Area} = \frac{180,000}{2} = 90,000\text{ sq ft}$.
• Step 3: Convert to acres: $\text{Acres} = \frac{90,000}{43,560} = 2.066\text{ Acres}$.

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4. Double Meridian Distance (DMD) Method Intro

Before computers, surveyors routinely computed polygon areas using the Double Meridian Distance (DMD) method derived from line Latitudes and Departures.

Rules of DMD Computation:

1. Rule 1: The DMD of the first course equals the Departure of the first course.
2. Rule 2: The DMD of any subsequent course equals the DMD of the preceding course + Departure of the preceding course + Departure of the course itself.
3. Rule 3: The DMD of the last course must equal the Departure of the last course with opposite sign.
4. Double Area: For each line, multiply its DMD by its Latitude.
5. Total Area: Sum the double areas, take the absolute value, and divide by 2.

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5. CST Exam Traps & Common Errors

[!WARNING] CST Level I Exam Traps:

1. Incorrect Acre Conversion Constant: Dividing square footage by 43,500 or 40,000 instead of 43,560 is a classic blunder. Memorize $43,560\text{ sq ft/acre}$!
2. Forgetting to Repeat the Initial Point: In the coordinate area method, failing to repeat Point 1 at the bottom of the table leaves the final side open, producing a completely false area.
3. Forgetting to Divide by 2: Calculating the difference of cross-products gives Double Area. Forgetting to divide by 2 yields an area exactly twice as large as the true lot size.
4. Misinterpreting Oblique Triangles: Applying $\frac{1}{2} \times b \times h$ to an oblique triangle using a side length in place of the perpendicular height $h$ yields an incorrect result. Always use the SAS formula ($\frac{1}{2} a b \sin C$) or Heron's formula when perpendicular height is not directly provided.