CELE Surveying (Geomatics) — Area and Volume Computation (Earthworks)Revision Notes

Formulas

Exam Tips

• Set up a table with columns: vertex, x, y, (x_i · y_{i+1}), (x_{i+1} · y_i) — this organizes the shoelace computation and reduces arithmetic errors.
• Board problems often give four to six vertices; writing out the table row by row is the fastest error-free approach.
• If bearing-distance data is given, always compute latitudes and departures first, then use DMD or convert to coordinates.
• Quick check: for a right triangle with legs a and b, area = ab/2 — use this to verify simple cases.
• Problems may state coordinates in 'northing and easting' — northing corresponds to y, easting corresponds to x.

Key Points

• The Shoelace formula computes the area of any closed polygon given the Cartesian coordinates of its vertices in order (either clockwise or counter-clockwise).
• The formula is: A = (1/2) |Σ(x_i · y_{i+1} − x_{i+1} · y_i)| where the subscripts wrap around (i.e., vertex n+1 = vertex 1).
• Always close the polygon in the summation by returning to the first vertex.
• The absolute value is taken to ensure a positive area regardless of vertex traversal direction.
• The DMD (Double Meridian Distance) method, used with latitudes and departures from compass/traverse surveys, yields the same result as the shoelace formula but is expressed in terms of survey data.
• For the DMD method: DMD of first course = departure of first course; DMD of next course = DMD of previous course + departure of previous course + departure of current course; 2A = Σ(DMD × latitude); take absolute value and divide by 2.
• Units: if coordinates are in metres, area is in m²; divide by 10,000 for hectares.

Definitions

Section Title

1. Area by Coordinates (Shoelace / Double-Meridian-Distance Method)

Common Mistakes

• Forgetting to close the polygon — the last term must multiply x_n by y_1 and x_1 by y_n.
• Not taking the absolute value — the shoelace sum can be negative if vertices are listed clockwise.
• In the DMD method, using the latitude of the wrong course as the multiplier.
• Mixing up latitude (N-S) and departure (E-W) when computing from bearings.
• Not converting area from m² to hectares when the problem asks for area in hectares (divide by 10,000).

Formulas

Exam Tips

• Memorize the multiplier pattern for Simpson's: 1, 4, 2, 4, 2, ..., 4, 1 — the first and last are 1, odd positions are 4, even interior positions are 2.
• When the problem gives an even number of offsets (odd number of intervals), apply Simpson's to the first even-interval group and add the remaining trapezoid separately.
• Always verify: (number of offsets − 1) = number of intervals. Simpson's needs this to be even.
• Board problems often present 5 or 9 offsets (4 or 8 intervals) — these are perfect for Simpson's rule.
• Trapezoidal rule always underestimates area for a curve that is concave upward and overestimates for concave downward. Simpson's is more balanced.

Key Points

• These rules apply when a boundary is irregular (e.g., a riverbank, road side slope) and is described by perpendicular offsets measured at equal intervals from a baseline.
• Offsets are labelled y_0, y_1, y_2, ..., y_n measured at spacing h along the baseline.
• Trapezoidal Rule treats each strip as a trapezoid; it is always applicable regardless of the number of intervals.
• Simpson's One-Third Rule treats pairs of strips as parabolic segments; it requires an even number of intervals (i.e., an odd number of offsets: n must be even, so the count of offsets is n+1 which is odd).
• Simpson's rule is more accurate than the trapezoidal rule for the same number of intervals because it fits a second-degree curve to each pair of strips.
• If the number of intervals is odd (even number of offsets), apply Simpson's rule to all but the last strip, then add the last strip as a trapezoid.
• Both rules assume equal spacing h between offsets.

Definitions

Section Title

2. Area of Irregular Boundaries — Trapezoidal and Simpson's Rules

Common Mistakes

• Applying Simpson's rule when the number of intervals is odd (even number of offsets) — check: count the offsets, subtract 1 to get the number of intervals; this must be even for pure Simpson's rule.
• Forgetting to halve the end offsets in the Trapezoidal rule formula.
• Misidentifying which offsets are 'odd-indexed' (multiplied by 4) vs 'even-indexed' (multiplied by 2) in Simpson's rule — remember: y_0 and y_n are each multiplied by 1, then alternate 4-2-4-2.
• Confusing 'number of offsets' with 'number of intervals': n intervals requires n+1 offsets.
• Forgetting that h is the spacing between offsets, not the total baseline length.

Formulas

Exam Tips

• The board exam frequently gives A_1, A_m (or asks you to compute it), and A_2, then asks for both end-area and prismoidal volumes — solve both and compute the prismoidal correction.
• Remember: L/2 for end-area; L/6 for prismoidal. The coefficient 4 in the prismoidal formula applies to A_m only.
• When mid-section dimensions are given (not the area), compute A_m from the cross-section geometry first before applying the prismoidal formula.
• For the mass diagram: a rising curve indicates net cut; a falling curve indicates net fill. The highest and lowest points correspond to transitions from cut to fill or fill to cut.
• If the problem says 'prismoidal formula' or 'more accurate method', use V = (L/6)(A_1 + 4A_m + A_2). If it says 'average end-area' or 'end-area method', use V = (L/2)(A_1 + A_2).
• Always check units: areas in m², distance in m → volume in m³. Convert to cubic meters then to bank cubic meters (BCM) if shrink/swell is specified.

Key Points

• Earthwork volume computations are used in highway, dam, and canal design to estimate quantities of soil to be cut (excavated) or filled.
• Two cross-sections, separated by a distance L (the station interval), define a solid whose volume is to be computed.
• End-Area Method: assumes the volume is a prismoid and approximates it by averaging the two end cross-sectional areas.
• Prismoidal Method: more accurate; uses the mid-section area A_m at the midpoint of L in addition to the two end areas.
• The mid-section area A_m is computed from the actual dimensions at the midpoint of the segment — it is NOT the simple average of A_1 and A_2.
• The Prismoidal Correction (C_p) is the difference between the end-area volume and the prismoidal volume: C_p = V_{end} − V_{prism}. If C_p > 0, end-area overestimates.
• The mass diagram (mass-haul curve) plots cumulative algebraic sum of cut minus fill volumes along the project alignment to determine haul distances, borrow pits, and waste sites.
• In Philippine highway practice, a free-haul distance and overhaul limit are specified; the mass diagram identifies sections requiring overhaul payment.

Definitions

Section Title

3. Earthwork Volumes — End-Area and Prismoidal Methods

Common Mistakes

• Using A_m = (A_1 + A_2)/2 instead of computing the true mid-section area from the actual midpoint dimensions — this is the most critical prismoidal formula error.
• Applying the end-area formula and calling it the prismoidal volume — the two formulas are different; know which one the problem asks for.
• Forgetting to multiply by L/2 (end-area) or L/6 (prismoidal) — computing only A_1 + A_2 without applying the distance factor.
• In mass diagram problems, treating cut as negative and fill as positive (or vice versa) inconsistently — convention: cut is positive (material available), fill is negative (material needed).
• Not distinguishing between volume of cut and volume of fill when swell and shrinkage factors are involved — in some board problems, bank measure vs. compacted measure is specified.

Formulas

Exam Tips

• On the board exam, when both Trapezoidal and Simpson's answers are among the choices, the question typically asks for Simpson's — it is more accurate and more commonly tested.
• When the problem mentions 'prismoidal correction', you must compute both end-area and prismoidal volumes.
• Always write down the formula first, then substitute values — this systematic approach prevents formula-substitution errors under exam pressure.

Key Points

• Area by Coordinates: use when vertices (x, y) or northings and eastings are given — apply Shoelace/DMD.
• Trapezoidal Rule: use for any number of offsets at equal spacing — always applicable.
• Simpson's Rule: use when number of intervals is even (odd number of offsets) — more accurate than trapezoidal.
• End-Area Volume: always computable from two end cross-sections — quick but slightly overestimates for tapering solids.
• Prismoidal Volume: use when mid-section area or midpoint dimensions are given — more accurate.
• Prismoidal Correction = V_end − V_prism; for most highway sections, this is positive (end-area overestimates).
• Mass diagram: cumulative cut (+) and fill (−) plotted vs. station to plan haul economics.

Section Title

4. Summary of Key Formulas and Decision Framework

Common Mistakes

• Skipping the decision step — always verify applicability of Simpson's rule before using it.
• Mixing up which method is more accurate: Simpson's > Trapezoidal for irregular boundaries; Prismoidal > End-Area for earthwork volumes.