ROUTE SURVEYING

EARTHWORKS
 Intended Learning Outcomes
1. Identify the methods in
calculating the area and volume
of cross sectional area for
earthworks
 Intended Learning Outcomes

2. Determine the various types of

cross sectional areas used in
calculating areas and volumes
of earthworks
EARTHWORKS
 Earthworks

Excavation
 Earthworks

Embankment
 Earthworks

Volume
 Volume of Earthworks

𝐿
𝑉𝑒𝑛𝑑 = 𝐴1 + 𝐴2
2

End Area Method

 Volume of Earthworks

𝐿
𝑉𝑝 = 𝐴1 + 4𝐴𝑚 + 𝐴2
6

Prismoidal Formula
 Volume of Earthworks

𝑉𝑝 = 𝑉𝑒𝑛𝑑 − 𝐶𝑣

Prismoidal Formula
 Volume of Earthworks

𝐿
𝐶𝑣 = 𝑐2 − 𝑐1 𝑑2 − 𝑑1
12

Prismoidal Correction
 Earthworks

Area
 AREA
COMPUTATIONS
 Area Computations

Counting Squares
Method
 Area Computations

Geometric
Method
 Geometric Method

𝑏ℎ
𝐴=
2

Area of a Triangle
 Geometric Method

2𝐴 = 𝑏ℎ

Double Area of a Triangle

 Geometric Method

𝑏1 + 𝑏2
𝐴=ℎ
2

Area of a Trapezoid
 Geometric Method

2𝐴 = ℎ 𝑏1 + 𝑏2

Double Area of a Trapezoid

 EXAMPLE
#1
Geometric Method
 Geometric Method

5.7 x 4.0
= 22.8m 2

2A of Triangle AJK
 Geometric Method

5.2 x (5.7+6.0)
= 60.8m 2

2A of Triangle ABJ
 Geometric Method

9.3 x (5.2+4.2)
= 87.4m 2

2A of Trapezoid BCIJ
 Geometric Method

15.0 x (4.2+4.7)
= 133.5m 2

2A of Trapezoid CDHI
 Geometric Method

4.8 x (4.7+1.5)
= 29.8m 2

2A of Trapezoid DEGH
 Geometric Method

1.5 x 4.5
= 6.8m 2

2A of Triangle EFG
 Geometric Method

22.8 + 60.8 + 87.4 + 133.5

+ 29.8 + 6.8
= 341.1m 2

Total Double Area (2A)

 Geometric Method

341.1/2
= 170.5m 2

Area of the Cross Section

 Area Computations

Double Meridian Distance

Method
(DMD)
Double Meridian Distance (DMD)

elevations
(latitudes)
 Double Meridian Distance (DMD)

distance from centerline

(departures)
 EXAMPLE
#2
Double Meridian Distance (DMD)
 Double Meridian Distance (DMD)

if the path goes up, +lat.

if the path goes down, -lat.

Sign Convention for Latitudes

 Double Meridian Distance (DMD)

if the path goes right, +dep.

if the path goes left, -dep.

Sign Convention for Departure

Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
A–B +0.1 -5.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
A–B +0.1 -5.0
B–C -1.1 -15.0

Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0
E–F +1.5 +30.0
F–I -1.5 +30.0
I–H -1.8 +7.0
H–G -0.9 -22.0
G–A +0.8 -15.0
A–B +0.1 -5.0
B–C -1.1 -15.0
C–D +1.9 -14.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0 4.0
E–F +1.5 +30.0 38.0
F–I -1.5 +30.0 98.0
I–H -1.8 +7.0 135.0
H–G -0.9 -22.0 120.0
G–A +0.8 -15.0 83.0
A–B +0.1 -5.0 63.0
B–C -1.1 -15.0 43.0
C–D +1.9 -14.0 14.0
Total
Double Area:
Course Latitude Departure DMD Double Area (2A)
D–E +1.0 +4.0 4.0 4.0
E–F +1.5 +30.0 38.0 57.0
F–I -1.5 +30.0 98.0 -147.0
I–H -1.8 +7.0 135.0 -243.0
H–G -0.9 -22.0 120.0 -108.0
G–A +0.8 -15.0 83.0 66.4
A–B +0.1 -5.0 63.0 6.3
B–C -1.1 -15.0 43.0 -47.3
C–D +1.9 -14.0 14.0 26.6
Total 385.0
Double Area:
 Double Meridian Distance

385m /2
2

= 192.50m 2

Area of the Cross Section

TYPES OF CROSS
SECTIONS
Different Cross Sections of Land
 Area Computations

Three-Level
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
 EXAMPLE
#3
Three Level Section
 Example #3

The following data are

cross-section notes of the
ground which will be
excavated for a roadway:
Three Level Section
 Example #3

Station 4+120
8.00 0 9.20
+2.00 +3.20 +2.80

Three Level Section

 Example #3

Station 4+160
9.50 0 10.70
+3.00 +2.60 +3.80

Three Level Section

 Example #3

the base of the road is 10m

and the side slopes are
1.5:1

Three Level Section

 Example #3
a.
find the volume of excavation
by end area method

Five Level Section

 Example #3

𝑐1 𝑏
𝐴𝑟𝑒𝑎1 = 8 + 9.2 + 2 + 2.8
2 2

Three Level Section

 Example #3

3.2 10
𝐴𝑟𝑒𝑎1 = 8 + 9.2 + 2 + 2.8
2 2

Three Level Section

 Example #3

3.2 10
𝐴𝑟𝑒𝑎1 = 8 + 9.2 + 2 + 2.8
2 2
𝟐
𝐴𝑟𝑒𝑎1 = 𝟑𝟗. 𝟓𝟐𝒎

Three Level Section

 Example #3

2.6 10
𝐴𝑟𝑒𝑎2 = 9.5 + 10.7 + 3 + 3.8
2 2

Three Level Section

 Example #3

2.6 10
𝐴𝑟𝑒𝑎2 = 9.5 + 10.7 + 3 + 3.8
2 2
𝟐
𝐴𝑟𝑒𝑎2 = 𝟒𝟑. 𝟐𝟔𝒎

Three Level Section

 Example #3

40
𝑉𝑒𝑛𝑑 = 39.52 + 43.26
2
𝟑
𝑉𝑒𝑛𝑑 = 𝟏𝟔𝟓𝟓. 𝟔𝒎

Three Level Section

 Example #3
b.
compute the volume by
prismoidal formula

Five Level Section

 Example #3
Station 4+120
8.00 0 9.20
+2.00 +3.20 +2.80

Station 4+160
9.50 0 10.70
+3.00 +2.60 +3.80
Three Level Section
 Example #3
Station 4+140
Horizontal Distance
8.00+9.50 0+0 9.20+10.70
2 2 2
Vertical Distance
2.00+3.00 3.20+2.60 2.80+3.80
2 2 2
Three Level Section
 Example #3

Station 4+140
8.75 0 9.95
+2.50 +2.90 +3.30

Three Level Section

 Example #3

2.9 10
𝐴𝑟𝑒𝑎𝑚 = 8.75 + 9.95 + 2.5 + 3.3
2 2

Three Level Section

 Example #3

2.9 10
𝐴𝑟𝑒𝑎𝑚 = 8.75 + 9.95 + 2.5 + 3.3
2 2
𝟐
𝐴𝑟𝑒𝑎𝑚 = 𝟒𝟏. 𝟔𝟐𝒎

Three Level Section

 Example #3

𝐿
𝑉𝑝 = 𝐴1 + 4𝐴𝑚 + 𝐴2
2

Three Level Section

 Example #3

40
𝑉𝑝 = 39.52 + 4 41.62 + 43.26
2

Three Level Section

 Example #3

40
𝑉𝑝 = 39.52 + 4 41.62 + 43.26
2
𝟑
𝑉𝑒𝑛𝑑 = 𝟏𝟔𝟔𝟏. 𝟕𝟑𝒎

Three Level Section

 Example #3
c.
determine the prismoidal
correction

Five Level Section

 Example #3

𝐿
𝐶𝑣 = 𝑐2 − 𝑐1 𝑑2 − 𝑑1
12

Three Level Section

 Example #3
𝑑1 = 8 + 9.20 = 17.2𝑚

Three Level Section

 Example #3
𝑑1 = 8 + 9.20 = 17.2𝑚
𝑑2 = 9.5 + 10.7 = 20.2𝑚

Three Level Section

 Example #3
𝑑1 = 8 + 9.20 = 17.2𝑚
𝑑2 = 9.5 + 10.7 = 20.2𝑚

40
𝐶𝑣 = 2.6 − 3.2 20.2 − 17.2
12

Three Level Section

 Example #3
𝑑1 = 8 + 9.20 = 17.2𝑚
𝑑2 = 9.5 + 10.7 = 20.2𝑚

40
𝐶𝑣 = 2.6 − 3.2 20.2 − 17.2
12
𝐶𝑣 = −𝟔𝒎𝟑
Three Level Section
 Area Computations

Special Case
Three-Level Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
 Area Computations

Five-Level
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
 EXAMPLE
#4
Five Level Section
 Example #4

it is required to determine the

earthwork volume on a portion
of road construction grading
work based on the following
cross-section notes:
Five Level Section
 Example #4

Station 7+460
+0.32 +0.60 +1.92 +1.52 +1.32
4.14 3.50 0 3.50 6.14

Five Level Section

 Example #4

Station 7+500
2.70 3.20 1.60 2.0 2.40
8.90 3.50 0 3.50 8.30

Five Level Section

Example #4
 Example #4
a.
determine the area of the first
section

Five Level Section

 Example #4

Area1
1 −3.5 −4.14 −3.5 0 3.5 6.14 3.5
2 0 0.32 0.60 1.92 1.52 1.32 0

Five Level Section

 Example #4
Area1
−3.5𝑥0.32 + (−4.14𝑥0.6)
+ −3.5𝑥1.92 + (3.5𝑥1.32)
1
2
− 0.32𝑥 − 3.5 − (1.92𝑥3.5)
− 1.52𝑥6.14 − (1.32𝑥3.5)
Five Level Section
 Example #4

Area1 = 12.63m 2

Five Level Section

 Example #4
b.
determine the area of the
second section

Five Level Section

 Example #4

Area2
1 −3.5 −8.9 −3.5 0 3.5 8.3 3.5
2 0 2.7 3.2 1.6 2 2.4 0

Five Level Section

 Example #4
Area2
−3.5𝑥2.7 + (−8.9𝑥3.2)
+ −3.5𝑥1.6 + (3.5𝑥2.4)
1
2
− 2.7𝑥 − 3.5 − (1.6𝑥3.5)
− 2𝑥8.3 − (2.4𝑥3.5)
Five Level Section
 Example #4

Area1 = 28.14m 2

Five Level Section

 Example #4
c.
compute the volume of
excavation using end area
method

Five Level Section

 Example #4

𝐿
𝑉𝑒𝑛𝑑 = 𝐴1 + 𝐴2
2

Five Level Section

 Example #4

40𝑚
𝑉𝑒𝑛𝑑 = 12.63 + 28.14
2

Five Level Section

 Example #4

𝑉𝑒𝑛𝑑 = 𝟖𝟏𝟓. 𝟒𝟎𝒎𝟑

Five Level Section

 Area Computations

Irregular
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
 Area Computations

Levelled
Section
Different Cross Sections of Land
Different Cross Sections of Land
Different Cross Sections of Land
CUT AND FILL
SECTIONS
 EXAMPLE
#5
Cut and Fill Section
 Example #5

Given the following cross-sections:

Base for Cut = 9m Sideslope = 1:1

Base for Fill = 8m Sideslope = 1.5:1

Cut and Fill Section

 Example #5

Station 3+000
5.48 0 5.00
+0.98 +3.05 +0.50

Cut and Fill Section

 Example #5

Station 3+060
6.76 0 4.63
+0.98 −1.22 −0.42

Cut and Fill Section

 Example #5

the base of the road is 10m

and the side slopes are
1.5:1

Cut and Fill Section

 Example #5
a.
compute the volume of cut
using end area method

Cut and Fill Section

 Example #5

5.05 9
𝐴𝑟𝑒𝑎1 = 5.48 + 5 + 0.98 + 0.5
2 2

Cut and Fill Section

 Example #5

5.05 9
𝐴𝑟𝑒𝑎1 = 5.48 + 5 + 0.98 + 0.5
2 2
𝟐
𝐴𝑟𝑒𝑎1 = 𝟏𝟗. 𝟑𝟏𝒎

Cut and Fill Section

 Example #5

1.22 8
𝐴𝑟𝑒𝑎2 = 6.76 + 4.63 + 1.84 + 0.42
2 2

Cut and Fill Section

 Example #5

1.22 8
𝐴𝑟𝑒𝑎2 = 6.76 + 4.63 + 1.84 + 0.42
2 2
𝟐
𝐴𝑟𝑒𝑎2 = 𝟏𝟏. 𝟒𝟕𝒎

Cut and Fill Section

 Example #5

42.86
𝑉𝑐𝑢𝑡 = 19.31
2
𝟑
𝑉𝑐𝑢𝑡 = 𝟒𝟏𝟑. 𝟖𝟏𝒎

Cut and Fill Section

 Example #5
b.
find the volume of fill using
end area method

Cut and Fill Section

 Example #5

60 − 42.86
𝑉𝑓𝑖𝑙𝑙 = 0 + 11.47
2
𝟑
𝑉𝑓𝑖𝑙𝑙 = 𝟗𝟖. 𝟑𝟎𝒎

Cut and Fill Section

 EXAMPLE
#6
Cut and Fill Section
 Example #6

From station 0+200 with center

height of 1.4m in fill, the ground line
makes a uniform slope of +5% to
station 0+260 whose center height
is 2.8m in cut.
Cut and Fill Section
 Example #6

Assuming both sections to be

trapezoidal with a roadway of 10m
and side slope of 2:1

Cut and Fill Section

 Example #6
a.
compute the grade of the
finished roadway

Cut and Fill Section

 Example #6

1.20
𝐺𝑟𝑎𝑑𝑒𝑟𝑜𝑎𝑑 =
60

𝐺𝑟𝑎𝑑𝑒𝑟𝑜𝑎𝑑 = 𝟎. 𝟐𝟎

Cut and Fill Section

 Example #6
b.
how far from station 0+200
will the filling extend?

Cut and Fill Section

 Example #6

𝑥 60 − 𝑥
=
1.4 2.8

𝑥 = 𝟐𝟎𝒎

Cut and Fill Section

 Example #6
c.
what is the elevation of the
section at station 0+250?

Cut and Fill Section

 Example #6

𝑦 2.8
=
30 40

𝑦 = 𝟐. 𝟏𝒎

Cut and Fill Section

 BORROW PIT
COMPUTATION
 Borrow Pit

an area where material

(usually soil, gravel or sand)
has been dug for use at
another location
 Borrow Pit

𝐴
𝑉𝑝𝑖𝑡 = ෍ ℎ1 + 2 ෍ ℎ2 + 3 ෍ ℎ3 + 4 ෍ ℎ4
4

Volume of Borrow Pit

 Borrow Pit

Elevation (h1)
 Borrow Pit

Elevation (h2)
 Borrow Pit

Elevation (h3)
 Borrow Pit

Elevation (h4)
 Borrow Pit

𝐴
𝑉𝑝𝑖𝑡 = ෍ ℎ1 + 2 ෍ ℎ2 + 3 ෍ ℎ3 + 4 ෍ ℎ4
4

Volume of Borrow Pit

 EXAMPLE
#7
Borrow Pit
 Example #7

A 90m x 90m square lot is to be

divided into 9 square sections. The
following data are the elevations at
the corners of the square sections
on the ground surface of the lot.
Volume of Borrow Pit
 Example #7

If the ground is to be leveled at

elevation 5m, find the total volume
of earthworks to be excavated.

Volume of Borrow Pit

 Example #7
A = 3.3m B = 7.9m C = 10.8m D = 6.8m

E = 7.6m F = 9.2m G = 10.6m H = 8.6m

I = 7.2m J = 10.2m K = 9.4m L = 6.9m

M = 7.2m N = 6.2m O = 9.6m P = 8.9m

Volume of Borrow Pit

A B C D

E F G H

I J K L

M N O P
A = 3.3m B = 7.9m C = 10.8m D = 6.8m

E = 7.6m F = 9.2m G = 10.6m H = 8.6m

I = 7.2m J = 10.2m K = 9.4m L = 6.9m

M = 7.2m N = 6.2m O = 9.6m P = 8.9m

 Example #7

෍ ℎ1 = 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑡 𝐴 𝐷 𝑀 𝑃

Volume of Borrow Pit

 Example #7

෍ ℎ1 = 3.3 + 1.8 + 3.9 + 2.2

Volume of Borrow Pit

 Example #7

෍ ℎ1 = 11.2m

Volume of Borrow Pit

 Example #7

෍ ℎ2 = 𝑒𝑙𝑒𝑣. 𝑎𝑡 𝐵 𝐶 𝐻 𝐿 𝑂 𝑁 𝐼 𝐸

Volume of Borrow Pit

 Example #7

෍ ℎ2 = 2.9 + 5.8 + 3.6 + 1.9

+ 4.6 + 1.2 + 2.2 + 2.6

Volume of Borrow Pit

 Example #7

෍ ℎ2 = 24.8m

Volume of Borrow Pit

 Example #7

෍ ℎ3 = 𝑁𝑂𝑁𝐸

Volume of Borrow Pit

 Example #7

෍ ℎ4 = 𝑒𝑙𝑒𝑣𝑎𝑡𝑖𝑜𝑛𝑠 𝑎𝑡 𝐹 𝐺 𝐾 𝐽

Volume of Borrow Pit

 Example #7

෍ ℎ4 = 4.2 + 5.6 + 5.2 + 4.4

Volume of Borrow Pit

 Example #7

෍ ℎ4 = 19.4m

Volume of Borrow Pit

 Example #7

𝐴
𝑉𝑝𝑖𝑡 = ෍ ℎ1 + 2 ෍ ℎ2 + 3 ෍ ℎ3 + 4 ෍ ℎ4
4

Volume of Borrow Pit

 Example #7

2
30
𝑉𝑝𝑖𝑡 = 11.2 + 2 24.8 + 4 19.4
4

Volume of Borrow Pit

 Example #7

2
30
𝑉𝑝𝑖𝑡 = 11.2 + 2 24.8 + 4 19.4
4

𝑉𝑝𝑖𝑡 = 𝟑𝟏𝟏𝟒𝟎𝒎𝟑

Volume of Borrow Pit