GELE Surveying (Geomatics) — Area and Volume Computation (Earthworks)Cheat Sheet
Area and Volume Computation (Earthworks) cheat sheet for GELE aspirants. If you could only take one sheet of paper into your review session, this is what it would look like. Professional Regulation Commission (PRC) — Board of Geodetic Engineering's most-tested concepts, all in one place.
Exam context
The Geodetic Engineer Licensure Examination is conducted by Professional Regulation Commission (PRC) — Board of Geodetic Engineering and is scheduled for September 2026. The Surveying (Geomatics) subtest is marked as "Core" in the official pattern, and Area and Volume Computation (Earthworks) appears in position 4th of 9 in the GELE Surveying (Geomatics) review rotation. Passing mark: 70% weighted average, no sub-test below 50%. Recent GELE 2026 papers have drawn roughly a meaningful share of questions from this subject.
Area and Volume Computation (Earthworks) - Cheat Sheet
Your final-30-minutes revision companion for PRC Civil Engineer Licensure Exam. Covers all formulas, definitions, and exam-critical facts on area by coordinates, irregular boundaries, and earthwork volumes.
Sections
Formulas
Formula
A = ½|Σ(x_i × y_{i+1} − x_{i+1} × y_i)| for closed polygon vertices (x_i, y_i)
Meaning
x_i, y_i = coordinates of vertex i; A = area; summation loops from i=1 to n (last vertex links back to first)
Watch Out
MUST take absolute value of result. Common error: forgetting the ½ multiplier or not closing the polygon (x_n+1 = x_1, y_n+1 = y_1)
When To Use
Finding area of any closed polygon given survey coordinates; also called DMD (double meridian distance) method
Formula
A = ½|Σ x_i(y_{i+1} − y_{i−1})| (alternative form using latitude differences)
Meaning
x_i = easting; y_{i+1}, y_{i−1} = northings of next and previous vertices
Watch Out
Ensure circular indexing: when i=1, use y_n for y_{i−1}; when i=n, use y_1 for y_{i+1}. Indexing errors are common.
When To Use
When coordinates given in latitude–departure form; equivalent to standard shoelace
Section Title
Area by Coordinates (Shoelace Method)
Important Facts
- Shoelace formula works for ANY closed polygon (convex or concave)
- Result is always positive if vertices listed counterclockwise; may be negative if clockwise (take absolute value)
- Requires closure: last vertex must connect back to first (x_{n+1} = x_1, y_{n+1} = y_1)
- Coordinates must be in same units (m, ft, etc.) and same coordinate system (UTM, local grid)
- No rounding intermediate steps; accumulate full precision before final division by 2
Key Definitions
Term
Shoelace Formula
Example
Triangle (0,0), (4,0), (0,3): A = ½|0(0−3) + 4(3−0) + 0(0−0)| = ½(12) = 6 m²
Definition
Mathematical method to compute area of polygon from vertex coordinates by summing cross-products of consecutive x and y values.
Term
DMD (Double Meridian Distance)
Example
Common in RA 544 Licensed Surveyor practice; part of NSCP recommended methods
Definition
Surveying method equivalent to shoelace; uses latitude and departure to compute area; often used with field traverse data.
Diagrams To Know
- Closed polygon with labeled vertices and coordinate grid
- Cross-product visualization (x_i × y_{i+1} vs x_{i+1} × y_i)
- Counterclockwise vs clockwise vertex numbering effect on sign
Formulas
Formula
A = h × [(y_0 + y_n)/2 + y_1 + y_2 + ... + y_{n−1}]
Meaning
h = uniform spacing between offsets (m); y_0, y_1, ..., y_n = perpendicular offsets (m); n = number of intervals
Watch Out
Slightly over-estimates area. NOT suitable for very irregular boundaries or sharp curves. First and last offsets get weight ½; interior offsets get weight 1.
When To Use
Finding area under irregular curve or along property boundary with equally spaced perpendicular offsets; any number of intervals OK
Section Title
Trapezoidal Rule (Irregular Boundaries)
Important Facts
- Trapezoidal rule works with ANY number of intervals (no parity restriction)
- Offsets must be perpendicular to baseline; uniform spacing h is ESSENTIAL
- First and last offsets weighted at 50%; all interior offsets weighted at 100%
- Accuracy increases with more intervals and smoother curves
- Best for moderately irregular boundaries; less accurate than Simpson's for smooth curves
Key Definitions
Term
Trapezoidal Rule
Example
Offsets 2, 5, 8, 6, 3 m at h=10 m spacing: A = 10×[(2+3)/2 + 5 + 8 + 6] = 10×(2.5 + 19) = 215 m²
Definition
Numerical integration method: approximates area under curve as sum of trapezoids with uniform width h and varying heights (offsets).
Diagrams To Know
- Series of trapezoids under a curve with equal base h and varying heights
- Offset diagram showing baseline with perpendicular measurements
Formulas
Formula
A = (h/3) × [(y_0 + y_n) + 4(y_1 + y_3 + y_5 + ...) + 2(y_2 + y_4 + y_6 + ...)]
Meaning
h = uniform spacing; y_0, y_n = end offsets; y_odd = 1st, 3rd, 5th... offsets (coeff 4); y_even = 2nd, 4th, 6th... offsets (coeff 2)
Watch Out
CRITICAL: Requires EVEN number of intervals. If you have even number of offsets, Simpson's FAILS — use trapezoidal or split off last strip. Coefficient pattern: 1-4-2-4-2-...-4-1.
When To Use
ONLY when number of intervals is EVEN (i.e., odd number of offsets: 3, 5, 7, 9...). More accurate than trapezoidal for smooth curves.
Section Title
Simpson's One-Third Rule (Irregular Boundaries)
Important Facts
- Requires EVEN number of intervals (ODD number of offsets: 3, 5, 7, 9, 11...)
- If offsets = 6, 7, 8, 9, 10 (even count = 5), CANNOT use Simpson's directly; use trapezoidal on first 4, then add trapezoid or use Simpson's on proper subset
- Coefficient sequence: 1-4-2-4-2-4-...-4-1 (always ends in 1)
- More accurate than trapezoidal for parabolic/smooth boundaries
- Accuracy deteriorates with very sharp bends or high-frequency variation
Key Definitions
Term
Simpson's One-Third Rule
Example
5 offsets (4 intervals): 2, 5, 8, 6, 3 m at h=10 m: A = (10/3)×[(2+3) + 4(5+6) + 2(8)] = (10/3)×(5+44+16) = (10/3)×65 = 216.7 m²
Definition
Numerical integration approximating area under smooth curve using parabolic arcs; more accurate than trapezoidal for regular curves.
Diagrams To Know
- Parabolic arcs fitting through three consecutive offsets
- Coefficient pattern diagram (1-4-2-4-2-1 sequence)
Formulas
Formula
V = (L/2) × (A_1 + A_2)
Meaning
V = volume (m³); L = horizontal distance between cross-sections (m); A_1, A_2 = area of cross-sections (m²)
Watch Out
Over-estimates volume for non-uniform (non-prismatic) cross-sections. L must be perpendicular distance between sections, not slope distance. Assumes average area (A_1+A_2)/2 applies over entire length L.
When To Use
Quick estimate of earthwork volume (cut/fill) between two survey cross-sections; assumes linear interpolation between sections
Section Title
End-Area Method (Earthwork Volumes)
Important Facts
- Simple and fast; commonly used in preliminary design
- Tends to over-estimate volume (especially for materials with irregular cross-sections)
- Applies average-end-area principle: assumes true volume lies at average of end areas
- L is center-to-center or station-to-station distance (horizontal projection)
- Suitable for approximately prismatic (uniform slope) earthwork; poor for irregular terrain
Key Definitions
Term
Cross-Section Area
Example
Road embankment at sta 10: measured area = 20 m²; at sta 10+50: area = 30 m²
Definition
Area of excavation or fill at a given station; measured perpendicular to grade line or baseline.
Term
End-Area Method
Example
A_1=20 m², A_2=30 m², L=50 m: V = (50/2)(20+30) = 1250 m³
Definition
Volume calculation averaging two end cross-section areas and multiplying by spacing; suitable for rough estimates or uniform slopes.
Diagrams To Know
- Two cross-section profiles showing A_1 and A_2 separated by distance L
- Side view of averaged volume between stations
Formulas
Formula
V = (L/6) × (A_1 + 4A_m + A_2)
Meaning
V = volume (m³); L = distance between end sections (m); A_1, A_2 = area at stations 1 and 2; A_m = area at MIDPOINT station (L/2 from each end)
Watch Out
A_m is NOT the average of A_1 and A_2. A_m must be computed from cross-section at L/2 location. If A_m is not available, approximate using A_m ≈ (A_1 + A_2)/2 and apply prismoidal correction.
When To Use
Accurate volume calculation when true mid-section area A_m can be measured or computed; follows prismoidal solid geometry
Section Title
Prismoidal Method (Earthwork Volumes)
Important Facts
- More accurate than end-area method for non-uniform terrain
- Requires three cross-section measurements: stations 0, L/2, L
- Coefficient pattern: 1-4-1 (symmetric, sums to 6)
- For very short sections (L < 15 m) or very irregular terrain, prismoidal advantage minimal
- If only A_1 and A_2 available, apply prismoidal correction to end-area result
Key Definitions
Term
Prismoidal Solid
Example
Earthwork volume = (L/6)(A_1 + 4A_m + A_2) is exact for any prismoidal section
Definition
Geometric shape whose volume can be exactly computed using end sections and mid-section; includes prisms, pyramids, frustums.
Term
Mid-Section Area (A_m)
Example
Stations 0+00 (A_1=20 m²), 0+25 (A_m measured = 28 m²), 0+50 (A_2=30 m²): prismoidal vol = (50/6)(20+4×28+30) = 1483.3 m³
Definition
Area of cross-section measured exactly at the midpoint (L/2) between two end sections, not the arithmetic mean of A_1 and A_2.
Diagrams To Know
- Three cross-sections: stations 0, L/2, L with labeled areas A_1, A_m, A_2
- Prismoidal solid shape with parabolic side profile
- Comparison graph: end-area vs prismoidal volume
Reactions Or Equations
Note
More common form: Correction ≈ (L/12) × (A_1 − 2A_m + A_2) when curvature exists
Equation
Prismoidal Correction = (L/6)(A_1 + A_2) − (L/2)(A_1 + A_2) = −(L/6)(A_1 + A_2 − 2A_avg)
Conditions
Used to adjust end-area estimate toward prismoidal volume when A_m is not available; correction is usually negative (reduces over-estimate)
Formulas
Formula
Correction = (L/12) × (A_1 − 2A_m + A_2)
Meaning
L = station spacing (m); A_1, A_2 = end-section areas; A_m = true mid-section area; correction adjusts end-area toward prismoidal
Watch Out
Correction is usually NEGATIVE (subtractive), reducing end-area over-estimate. If (A_1 − 2A_m + A_2) is positive, subtract correction; if negative, add it. Common error: wrong sign.
When To Use
Converting end-area volume estimate to more accurate prismoidal value when mid-section A_m is known; correction is ADDITIVE or SUBTRACTIVE
Formula
V_prismoidal = V_end−area + Correction = (L/2)(A_1 + A_2) + (L/12)(A_1 − 2A_m + A_2)
Meaning
Direct formula combining end-area and correction into single step
Watch Out
Algebraically equivalent to V = (L/6)(A_1 + 4A_m + A_2), but often used in practice to adjust preliminary estimates
When To Use
One-step calculation of prismoidal volume from three sections without computing end-area first
Section Title
Prismoidal Correction
Important Facts
- Correction reflects curvature or variation in cross-section shape between end sections
- Positive correction (add to end-area) when middle section is LARGER than average of ends
- Negative correction (subtract from end-area) when middle section is SMALLER than average of ends
- Magnitude of correction proportional to (A_1 − 2A_m + A_2); zero if section uniform
- Standard practice in earthwork volume reports: state both end-area and corrected volumes
Key Definitions
Term
Prismoidal Correction
Example
End-area = 1250 m³; A_1=20, A_m=28, A_2=30, L=50: Correction = (50/12)(20−56+30) = −28.3 m³; V_prismoidal = 1250−28.3 = 1221.7 m³
Definition
Adjustment factor applied to end-area volume to account for non-prismatic shape; equal to (L/12)(A_1 − 2A_m + A_2).
Diagrams To Know
- Graphical representation of three cross-section areas on vertical axis vs station on horizontal
- Sign convention: upward curvature → positive correction; downward curvature → negative correction
Formulas
Formula
Cumulative Volume = Σ(Cut − Fill) from station 0 to current station
Meaning
Cut = excavation volume (positive); Fill = placement volume (negative); cumulative tracks net material balance
Watch Out
Mass diagram ordinate values depend on reference datum; changing datum shifts entire curve vertically. Slope changes at each new section indicate cut/fill transition.
When To Use
Planning material haul, identifying borrow/waste sites, balancing cut and fill on linear projects (roads, pipelines, canals)
Common Values
Value
100–300 m depending on project specification
Symbol
d_fh
Quantity
Free Haul Distance (typical)
Value
0.5–2 USD (varies by region and material)
Symbol
C_oh
Quantity
Cost of Over-Haul (per station-m³)
Section Title
Mass Diagram (Haul & Borrow/Waste Plan)
Important Facts
- Upward slope = cut section (material available)
- Downward slope = fill section (material needed)
- Horizontal section = balance (no net cut/fill)
- Minimum point on curve = maximum borrow required at that location
- Maximum point = maximum haul distance endpoint
- Straight-line connections between stations assume linear interpolation of volume change
- Area between diagram and zero-line × scale = actual haul (station-m³)
Key Definitions
Term
Mass Diagram
Example
Road section: sta 0 (cut=50 m³), sta 1 (fill=−30 m³), sta 2 (cut=40 m³). Cumulative: 0→50→20→60 m³
Definition
Graph of cumulative (cut − fill) volume vs station; used to plan haul distances, borrow areas, and waste disposal for earthwork balancing.
Term
Haul Distance
Example
Mass diagram area/height ratio gives weighted haul distance in station-meters
Definition
Average distance material is transported from cut site to fill site; derived from mass diagram using geometric center method.
Term
Borrow Pit
Example
Station 3 cumulative mass = −20 m³ (below zero line) → 20 m³ borrow required
Definition
External source of fill material when project cut volume is insufficient; identified where mass diagram goes negative and cannot be filled internally.
Diagrams To Know
- Mass diagram: cumulative volume (y-axis) vs station (x-axis); upslope = cut, downslope = fill
- Free-haul and over-haul diagram (derivative of mass curve showing haul distances)
- Cut/fill balance line (equilibrium line) showing optimal material distribution
Section Title
Volume Calculation Decision Tree
Important Facts
- Choose method based on available data and required accuracy
- End-area: fastest, works with 2 sections, over-estimates ~5–10%
- Prismoidal: requires 3 sections, more accurate, standard for final estimates
- Simpson's rule: for irregular boundary area, requires EVEN intervals
- Trapezoidal: any number of offsets, simpler than Simpson's, slightly less accurate
Diagrams To Know
- Decision flow: given data → suitable volume method
Must Remember
- 1. SHOELACE FORMULA: A = ½|Σ(x_i·y_{i+1} − x_{i+1}·y_i)| — MUST close polygon and take absolute value. Most direct method for coordinates.
- 2. SIMPSON'S INTERVAL REQUIREMENT: EVEN number of intervals ONLY (= ODD number of offsets: 3, 5, 7, 9...). If wrong parity, use trapezoidal or split the section.
- 3. SIMPSON'S COEFFICIENT PATTERN: Always 1-4-2-4-2-...-4-1 (ends in 1, not 4). Common exam trap: students reverse or forget pattern.
- 4. MID-SECTION AREA (A_m): In prismoidal V = (L/6)(A_1 + 4A_m + A_2), the A_m is measured AT station L/2, NOT the arithmetic average of A_1 and A_2. This is the #1 mistake.
- 5. END-AREA OVER-ESTIMATES: V = (L/2)(A_1 + A_2) gives fast answer but typically 5–10% high. Apply prismoidal correction (L/12)(A_1 − 2A_m + A_2) for accuracy.
- 6. PRISMOIDAL VS END-AREA: Prismoidal is MORE accurate (error < 1% for prismoids). Use it for final estimates, design submissions, and RA 544 surveys; end-area for preliminary checks.
- 7. UNIFORM SPACING REQUIRED: Trapezoidal and Simpson's both need uniform offset spacing h. Non-uniform spacing invalidates these methods; must split into uniform sections.
- 8. TRAPEZOIDAL WEIGHTS: First offset ×½, all interior offsets ×1, last offset ×½. Common error: treating all as ×1.
- 9. MASS DIAGRAM INTERPRETATION: Upward slope = cut (excess material), downward = fill (shortage). Minimum point = max borrow needed. Used for RA 544 haul planning.
- 10. FORMULA SELECTION DECISION: Given coordinates → Shoelace. Given perpendicular offsets → Simpson's (if even intervals) or Trapezoidal. Given cross-sections → End-area (quick) or Prismoidal (accurate). Always verify data type before applying formula.
Last Minute Tips
- CHECK PARITY FIRST: Before writing Simpson's formula, count offsets. If EVEN count (not odd), use trapezoidal or split. Mismatched parity = automatic error.
- ABSOLUTE VALUE IN SHOELACE: Always take |result| at the end, even if you think the polygon is oriented correctly. Exam setters often reverse vertex order to test this.
- THREE SECTIONS FOR PRISMOIDAL: If problem gives only A_1 and A_2, you MUST use end-area. Don't invent A_m; instead, state that prismoidal requires mid-section data. Partial credit for recognizing the limitation.
- MASS DIAGRAM SIGN CONVENTION: Plot (Cut − Fill), not (Fill − Cut). Positive = surplus excavation, negative = shortfall. Check diagram slope direction matches cut/fill phase of project.
- UNITS CONSISTENCY: All offsets/coordinates in same units (m, not mixed m and cm). All areas in m² or unit². Volume in m³. Common exam trap: mixing units mid-calculation.
Comparison Tables
Rows
Values
- A = ½|Σ(x_i×y_{i+1} − x_{i+1}×y_i)|
- Any closed polygon
- Exact (no approximation)
- Exact
- Any parcel with known coordinates; property survey
- Forgetting absolute value; incorrect indexing (not closing polygon)
Property
Shoelace (Coordinates)
Values
- A = h[(y_0+y_n)/2 + Σy_interior]
- Any number ≥ 2
- Moderate (~1–3% error)
- Approximate
- Irregular boundary with equally spaced offsets; any count OK
- Spacing not uniform; first/last coefficients = 1 instead of 0.5
Property
Trapezoidal
Values
- A = (h/3)[(y_0+y_n) + 4Σodd + 2Σeven]
- EVEN number only (odd count)
- High (~0.1–0.5% error for smooth curves)
- Approximate (parabolic)
- Smooth irregular boundary; higher accuracy needed
- Using with ODD number of intervals; wrong coefficient pattern (1-4-2-...)
Property
Simpson's One-Third
Columns
- Method
- Formula
- Number of Offsets
- Accuracy
- Best Use Case
- Common Error
Table Title
Area Calculation Methods Comparison
Rows
Values
- V = (L/2)(A_1 + A_2)
- 2 (end sections)
- Low–Moderate
- Over-estimates 5–10%
- Preliminary design; quick estimates; uniform terrain
Property
End-Area
Values
- V = (L/6)(A_1 + 4A_m + A_2)
- 3 (end + mid)
- High
- Error < 1% for true prismoids
- Final estimates; regulatory submissions; non-uniform sections; RA 544 cadastral surveys
Property
Prismoidal
Values
- V = (L/2)(A_1+A_2) + (L/12)(A_1−2A_m+A_2)
- 3 (end + mid)
- High (same as prismoidal)
- Error < 1%
- Adjusting end-area preliminary estimate when A_m becomes available
Property
Prismoidal Correction
Columns
- Method
- Formula
- Sections Required
- Accuracy Level
- Typical Error
- When to Use
Table Title
Volume Calculation Methods Comparison
Rows
Values
- y_0
- 0
- 1
- 1
- y_0 × 1
Property
First offset
Values
- y_1
- 1
- 4
- 4
- y_1 × 4
Property
2nd offset (odd)
Values
- y_2
- 2
- 2
- 2
- y_2 × 2
Property
3rd offset (even)
Values
- y_3
- 3
- 4
- 4
- y_3 × 4
Property
4th offset (odd)
Values
- y_4
- 4
- 1
- 1
- y_4 × 1
Property
Last offset (even)
Columns
- Offset Position
- Index
- Coefficient
- Weight Factor
- Example (5 offsets)
Table Title
Simpson's One-Third Rule: Coefficient Application
Rows
Values
- 2 cross-sections 50 m apart
- End-area
- V = (50/2)(A_1 + A_2)
- A_1=20, A_2=30 m² → V=1250 m³
Property
Highway fill section
Values
- 3 cross-sections: sta 0, 0+25, 0+50
- Prismoidal (if A_m measured)
- V = (L/6)(A_1 + 4A_m + A_2)
- A_1=20, A_m=28, A_2=30, L=50 → V=1483 m³
Property
Channel excavation (final design)
Values
- Perpendicular offsets at 10 m spacing
- Simpson's or Trapezoidal
- A = (h/3)[(y_0+y_n)+4Σodd+2Σeven]
- 5 offsets: 2, 5, 8, 6, 3 m → A=216.7 m²
Property
Irregular boundary (land survey)
Columns
- Scenario
- Data Available
- Best Method
- Formula to Use
- Example
Table Title
Cross-Section Measurement & Volume: Common Setups
Previous chapter
Traverse and Omitted Measurements
Next chapter
Horizontal Curves (Simple, Compound, Reverse)
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