LEC8 Earthwork Area Notes

Topic 8: Earthwork - Area

CM 1331: Construction Surveying, Spring 2025, Instructor: Dr. Farah Naz

Why Determine Areas?

• Reasons:
  • Acreage of land in property deeds.
  • Acreage of fields and lakes.
  • Quantity of surface area for paving, seeding, or sodding (in square yards).
  • End areas for earthwork volume calculation.
• In plane surveying:
  • Area is the orthogonal projection of the surface onto a horizontal plane.

Methods of Measuring Area

• Both field and map measurements are used.
• Field measurements are generally more accurate.
• Field measurement methods:
  • Division of the tract into simple figures (triangles, rectangles, trapezoids).
  • Offsets from a straight line.
  • Coordinates.
  • Double-meridian distances.
• Methods of determining area from map measurements:
  • Counting coordinate squares.
  • Dividing the area into triangles, rectangles, or other regular geometric shapes.
  • Digitizing coordinates.
  • Using a planimeter over the enclosing lines.
• Area determination invariably depends on field observations.

Approximate Area Conversion Factors

• $1 \text{ meter} = 3.28084 \text{ ft} = 39.3701 \text{ inch}$
• $1 \text{ yd} = 3 \text{ ft}$
• $1 \text{ hectare} = 10000 \text{ square meter}$
• $1 \text{ acre} = 43560 \text{ square foot}$

Area by Division into Simple Figures

• A tract can be divided into geometric figures like triangles, rectangles, or trapezoids.
• Sides and angles of these figures can be observed in the field.
• Individual areas are calculated and totaled.

Area by Offsets from Straight Lines

• Irregular tracts can be reduced to a series of trapezoids.
• This is done by observing right-angle offsets from points along a reference line.
• The reference line is marked by stationing, and offset positions are given by stations and pluses.
• Spacing between offsets may be regular or irregular, depending on conditions.

Area by Offsets from Straight Lines - Regularly Spaced Intervals

• If offsets are at regularly spaced intervals, area can be found by a formula.
• $b$ is the length of a common interval between offsets.
• $h<em>0, h</em>1, h_2, \dots$ are the offsets.
• The regular interval for the example is a half-station, or 50 ft.
• (Math problem solved in class)

Irregularly Spaced Offsets

• For irregularly curved boundaries, the spacing of offsets along the reference line varies.
• Spacing should accurately define the curved boundary when adjacent offset points are connected by straight lines.
• A formula is used for calculating area in this case.
• $a, b, c, \dots$ are the varying offset spaces.
• $h<em>0, h</em>1, h_2, \dots$ are the observed offsets.
• (Math problem solved in class)

Area Computation by the Coordinate Method

• Computation of area within a closed polygon is commonly done by the coordinate method.
• Coordinates of each angle point in the figure must be known.
• Two formulas are available.

Area Computation by the Coordinate Method - Important Considerations

• Parcels, tracts, and polygons can have different shapes.
• The formula for the area of polygons changes with their shapes.
• A specific formula is given for the polygon ABCDEA as shown in a figure.
• Sort the coordinates in counterclockwise direction to apply the correct formula and calculate the correct area.

Example Calculation of Area by Coordinate Method

• Calculate the area by coordinate method using provided coordinates for polygon ABCDEA.
• Point A: X = 0.00, Y = 591.78
• Point B: X = 517.44, Y = 202.94
• Point C: X = 523.41, Y = 0.00
• Point D: X = 716.29, Y = 694.02
• Point E: X = 125.72, Y = 847.71

Area of Polygon by Coordinates - Calculation and Results

• Double Area (ft²) = $1,044,861 - 499,684 = 545,177$
• Area = $545,177 / 2 = 272,588 \text{ ft}^2$
• Area = $6.258$ acres

References

• Chapter 12, Elementary Surveying: An Introduction to Geomatics, 16th edition, 2022. Author: Charles D Ghilani.