Earthwork formulas

Surveying · FE Reference Handbook section

Surveying

5 formulas

10 exam-style examples

~55 min

All Surveying lectures

Learning objectives

What you must be able to do before leaving this section.

This chapter section covers Earthwork formulas within Surveying. Read it the way you would read a textbook chapter: the theory first so the relations mean something, then every equation with its use and its trap, then 10 fully worked examples with the arithmetic shown line by line, and finally a self-check you should be able to answer without notes.

Explain, in your own words, what earthwork formulas describes physically and when it applies.
State every one of the 5 relations the handbook lists here and name each symbol with its unit.
Select the correct relation from the wording of an exam stem within 20 seconds.
Carry a complete solution from givens to a "most nearly" answer with the correct unit.
Recognise the distractors generated by the unit trap: bearings in quadrant form, azimuths from north.

Lecture

Why this section exists. Earthwork formulas is the part of Surveying that lets you connect a closed traverse or a cut-and-fill section to a number you can defend. Before any equation is useful you must be able to picture the physical situation it describes; the schematic below is that picture.

How the theory is built. The handbook prints results, not derivations. Each relation in this section comes from one governing principle applied to the idealised system: state the principle, impose the stated assumptions, and the printed equation follows. Knowing which assumption each relation rests on is what lets you reject a wrong answer choice in seconds.

How it is examined. Items from this page are written as a closure, an area, or an earthwork volume. Roughly two thirds are direct substitution, one third require one intermediate quantity from a neighbouring relation, and a small number are conceptual — testing whether you know the assumption, not the arithmetic.

The habit that earns the points. Unit discipline. bearings in quadrant form, azimuths from north. Every relation below is dimensionally consistent only when that rule is honoured, and the distractor set is deliberately built from candidates who ignored it. Write the unit next to every number you substitute, every time.

How to study this page. Read the theory, then cover the formula cards and try to reproduce each relation from its description. Then work the examples with the solution hidden, revealing one line at a time. Finish with the self-check questions; if you cannot answer one, return to the matching formula card.

Surveyor operating a total station beside a rural highway with a rod person downstream. — Photo 1. Where this shows up in practice: earthwork formulas.
Capstone Studio instructional photograph

Surveying — Earthwork formulas: reference schematic for orienting the symbols used in this section.

Theory, developed

Read this before the equations — it is what makes them memorable.

The physical situation

Every item from this section describes a closed traverse or a cut-and-fill section. Sketch it before you compute — a labelled sketch with the givens on it converts a wordy stem into a solvable problem and exposes the quantity the examiner left out on purpose.

The governing principle

The 5 relations on this page are consequences of one principle applied to that idealised system. Identify which quantity is conserved, balanced, or defined, and the correct equation follows without memorisation.

Assumptions and limits of validity

Each printed relation carries silent assumptions — linearity, steady state, uniformity, small deformation, or standard conditions, depending on the subject. Conceptual exam items are written by violating exactly one of these, so read the sentence above the equation as carefully as the equation itself.

Solution procedure you should automate

1) Read the last sentence of the stem to identify the requested quantity. 2) Locate the relation on this page whose left-hand side is that quantity. 3) Tabulate the givens with units and mark the missing symbol. 4) If a symbol is missing, find the one relation that produces it. 5) Rearrange symbolically, substitute once, evaluate, and round only at the end.

Notation used in this section

V	Quantity produced by "V = L(A1 + A2)/2" — read its definition and unit from the handbook line directly above the equation.
Am	Quantity produced by "Am = area of mid-section" — read its definition and unit from the handbook line directly above the equation.
L	Quantity produced by "L = distance between A1 and A2" — read its definition and unit from the handbook line directly above the equation.

Handbook notes for this section

Definitions and conditions exactly as the handbook states them.

Average End Area Formula
Prismoidal Formula
where
Pyramid or Cone

Core formulas for this FE topic

Definitions, applicability, units, assumptions and worked examples for each relation.

Worked exam-style examples

The four ways this section is written on the real exam — thoughts first, then equations, then substitution.

Example 1

Earthwork volume by average end area

Two cross sections 30 m apart have cut areas of 24.5 m² and 31.7 m². Compute the volume between them, then the loose volume with 12% swell.

Given

A₁ = 24.5 m², A₂ = 31.7 m²
L = 30 m
Swell = 12%

Find

Bank volume and loose (hauled) volume

Start with the thinking

Average end area is the FE default unless the prismoidal method is requested.
Swell increases hauled volume relative to bank volume.

Step-by-step solution

Average area
$(24.5 + 31.7) /2 = 28.1 m^{2}$
Bank volume
$V = 28.1 (30) = 843 m^{3}$
Swell factor
$1 + 0.12 = 1.12$
Loose volume — 843(1.12)
Result
$V_{b} ank = 843 m^{3}, V_{l} oose = 944 m^{3}$

Answer: 843 m³ bank, 944 m³ loose

Why the other options are there

1,686 m³ (areas added, not averaged)
753 m³ (shrinkage applied instead of swell)

Reference: FE Reference Handbook — Construction — Earthwork volumes

Example 2

Earthwork volume by average end area — Earthwork formulas

Two stations 65 ft apart have cut areas of 178 ft² and 126 ft². What is the volume between them in cubic yards?

Given

A₁ = 178 ft²
A₂ = 126 ft²
L = 65 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 65 (178 + 126) /2 = 9, 880 f t^{3}$
Convert
$V = 9, 880/27 = 365.9 y d^{3}$

Answer: ≈ 365.9 yd³

Why the other options are there

9,880 yd³ (units not converted)
731.9 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 3

Earthwork volume by average end area — Earthwork formulas (2)

Two stations 68 ft apart have cut areas of 87 ft² and 96 ft². What is the volume between them in cubic yards?

Given

A₁ = 87 ft²
A₂ = 96 ft²
L = 68 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 68 (87 + 96) /2 = 6, 222 f t^{3}$
Convert
$V = 6, 222/27 = 230.4 y d^{3}$

Answer: ≈ 230.4 yd³

Why the other options are there

6,222 yd³ (units not converted)
460.9 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 4

Earthwork volume by average end area — Earthwork formulas (3)

Two stations 110 ft apart have cut areas of 60 ft² and 162 ft². What is the volume between them in cubic yards?

Given

A₁ = 60 ft²
A₂ = 162 ft²
L = 110 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 110 (60 + 162) /2 = 12, 210 f t^{3}$
Convert
$V = 12, 210/27 = 452.2 y d^{3}$

Answer: ≈ 452.2 yd³

Why the other options are there

12,210 yd³ (units not converted)
904.4 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 5

Earthwork volume by average end area — Earthwork formulas (4)

Two stations 56 ft apart have cut areas of 71 ft² and 80 ft². What is the volume between them in cubic yards?

Given

A₁ = 71 ft²
A₂ = 80 ft²
L = 56 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 56 (71 + 80) /2 = 4, 228 f t^{3}$
Convert
$V = 4, 228/27 = 156.6 y d^{3}$

Answer: ≈ 156.6 yd³

Why the other options are there

4,228 yd³ (units not converted)
313.2 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 6

Earthwork volume by average end area — Earthwork formulas (5)

Two stations 182 ft apart have cut areas of 85 ft² and 125 ft². What is the volume between them in cubic yards?

Given

A₁ = 85 ft²
A₂ = 125 ft²
L = 182 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 182 (85 + 125) /2 = 19, 110 f t^{3}$
Convert
$V = 19, 110/27 = 707.8 y d^{3}$

Answer: ≈ 707.8 yd³

Why the other options are there

19,110 yd³ (units not converted)
1,416 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 7

Earthwork volume by average end area — Earthwork formulas (6)

Two stations 88 ft apart have cut areas of 50 ft² and 103 ft². What is the volume between them in cubic yards?

Given

A₁ = 50 ft²
A₂ = 103 ft²
L = 88 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 88 (50 + 103) /2 = 6, 732 f t^{3}$
Convert
$V = 6, 732/27 = 249.3 y d^{3}$

Answer: ≈ 249.3 yd³

Why the other options are there

6,732 yd³ (units not converted)
498.7 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 8

Earthwork volume by average end area — Earthwork formulas (7)

Two stations 108 ft apart have cut areas of 173 ft² and 107 ft². What is the volume between them in cubic yards?

Given

A₁ = 173 ft²
A₂ = 107 ft²
L = 108 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 108 (173 + 107) /2 = 15, 120 f t^{3}$
Convert
$V = 15, 120/27 = 560.0 y d^{3}$

Answer: ≈ 560.0 yd³

Why the other options are there

15,120 yd³ (units not converted)
1,120 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 9

Earthwork volume by average end area — Earthwork formulas (8)

Two stations 128 ft apart have cut areas of 151 ft² and 164 ft². What is the volume between them in cubic yards?

Given

A₁ = 151 ft²
A₂ = 164 ft²
L = 128 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 128 (151 + 164) /2 = 20, 160 f t^{3}$
Convert
$V = 20, 160/27 = 746.7 y d^{3}$

Answer: ≈ 746.7 yd³

Why the other options are there

20,160 yd³ (units not converted)
1,493 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Example 10

Earthwork volume by average end area — Earthwork formulas (9)

Two stations 107 ft apart have cut areas of 91 ft² and 118 ft². What is the volume between them in cubic yards?

Given

A₁ = 91 ft²
A₂ = 118 ft²
L = 107 ft
27 ft³ = 1 yd³

Find

Volume in yd³

Start with the thinking

Average end area is the FE default unless the prismoidal formula is requested.
Convert only at the end.

Step-by-step solution

Average end area
$V = L (A_{1} + A_{2}) /2$
Substituting
$V = 107 (91 + 118) /2 = 11, 182 f t^{3}$
Convert
$V = 11, 182/27 = 414.1 y d^{3}$

Answer: ≈ 414.1 yd³

Why the other options are there

11,182 yd³ (units not converted)
828.3 yd³ (average not taken)

Reference: FE Reference Handbook — Surveying → Earthwork formulas

Self-check

Answer these without notes before moving on.

Without looking, state the relation on this page whose left-hand side is the quantity most often requested, and name every symbol in it.
Which assumption, if violated, makes the main relation of this section invalid?
Given a closed traverse or a cut-and-fill section, what is the first quantity you would compute, and why that one first?
Which unit conversion in this subject most often produces a wrong answer choice, and what is its numerical factor?
Rework Example 1 above from the givens alone, without reading the solution lines.

Chapter summary

Earthwork formulas contains 5 relations; you must be able to find this page in under 15 seconds.
Exam style: a closure, an area, or an earthwork volume.
Unit rule: bearings in quadrant form, azimuths from north.
Work the 10 examples until the solution path, not the answer, is automatic.

Common traps in this section

bearings in quadrant form, azimuths from north
Answering the intermediate quantity instead of the quantity requested.
Rounding intermediate values before the final step.
Using a relation from an adjacent handbook section that shares a symbol.
Skipping the sketch — most lost points on this page start with a misread geometry.