Introduction, computation of area, computation of area from field notes and plotted plans, boundary area, area of traverse, Use of Plannimeter, computations of volumes, Volume from cross sections, Trapezoidal and Prismoidal formulae, Prismoidal correction, Curvature correction, capacity of reservoir, volume from borrow pits.

1. 1
PREPARED BY : ASST. PROF. VATSAL D. PATEL
MAHATMA GANDHI INSTITUTE OF
TECHNICAL EDUCATION &
RESEARCH CENTRE, NAVSARI.

2.  The term ‘‘area’’ in the context of surveying refers to the area
of land projected upon the horizontal plane and not to the
actual area of the land surface.
 One of the primary objects of land survey is to determine the
area of field.
2

3.  Square metres
 Hectares ( 1 Hectare = 10000 m2 = 2.471 acres )
 Square feet
 Acres (1 acre = 4840 sq. yd. = 43560 sq. ft. )
 Square kilometre (km2) = (1 km2 = 106 m2 )
3

4. Computation of Area
Graphical method
From field notes From plotted plan
Entire area Boundary area
Mid-ordinate
rule
Average ordinate
rule
Trapezoidal
rule
Simpson’s
rule
Instrumental method
4

5.  In cross-staff survey, the area of field can be directly
calculated from field notes. During the survey work the whole
area is divided into some specific geometrical shapes like,
square, rectangular, triangle etc… and then the total area can
be computed directly.
5

6.  In this method, chain line is run approximately in the centre of
the area to be calculated with the help of the cross-staff or
optical square, perpendicular offsets are marked on the chain
line. Offset are marked along with chainage and then the area
is calculated by forming geometrical figures.
6

7.  The area can be calculated from the following two ways.
 Case 01 – Considering the Entire Area :
 The entire area is divided into regions of convenient shape and
the area is calculated as follows :
 By dividing the area into triangles
 By dividing the area into squares
 By drawing parallel lines and converting them to rectangles
7

8.  By dividing the area into triangles :
 Triangle area = ½ x base x altitude
 Area = Sum of areas of triangles
8
1
4
2
3

9.  By dividing the area into squares :
 Each square represent unit area 1 cm2 or 1 m2
 Area = Nos. of square x Unit area
9

10.  By drawing parallel lines and converting them to rectangles :
 Area = ∑ Length of rectangles x Constant depth
10

11.  Case 02 – Middle Area + Boundary Area :
 In this method, a large square or rectangle is formed within the
area in the plan. Then ordinates are drawn at regular intervals
from the sides of the square to the curved boundary. The
middle area is calculated in the usual way.
11

12.  Middle Area + Boundary Area :
 Total area A = Middle area A1 + Boundary area A2
12
A1
A2

13.  Boundary area is calculated according to one of the following
rules :
 The mid-ordinate rule
 The average ordinate rule
 The trapezoidal rule
 Simpson’s rule
 Coordinate method
13

14.  Refer the figure,
14

15.  Let, O1 , O2 , O3 ......., On = Ordinates at equal intervals
 l = length of Base Line
 d = Common Distance between ordinates
 h1 , h2 , h3, ........, hn = Mid-Ordinates
 Area of plot = h1 x d + h2 x d +..... + hn x d
= d (h1 + h2 +..... + hn )
 Area = Common distance x Sum of mid-ordinates
15

16.  Refer the figure,
16

17.  Let, O1 , O2 ,....,On = Ordinates or Offsets at regular intervals
 l = length of Base Line
 n = Number of divisions
 n +1 = Number of ordinates
17

18.  While applying the trapezoidal rule, boundaries between the
ends of the ordinates are assumed to be straight.
 Thus, the area enclosed between the base line and the irregular
boundary line are considered as trapezoids.
 Refer the figure,
18

19.  Let, O1 , O2 ,.....,On = Ordinates at equal intervals
 d = common distance
19

20.  Total area = (O1 + 2O2 + 2O3 + 2O4+ 2On-1 + On) x (d/2)
= (O1 + On + 2(O2 + O3 +…+ On-1) x (d/2)
= (Common distance/2) x [(1st ordinate +last
ordinate) + 2 (sum of other ordinates)]
 Limitation :
 There is no limitation for this rule. This rule can be applied for
any number of ordinates.
20

21.  In this rule, the boundaries between the ends of ordinates are
assumed to form an arc of a parabola. Hence Simpson’s Rule
is sometimes called the parabolic rule.
 Refer the figure,
21
1 2 3

22.  Let, O1 , O2 , O3 = three consecutive ordinates
 d = common distance
 Area of AF2DC = Area of trapezium AFDC + Area of
segment F2DEF
 Here, Area of trapezium = x 2d
22
2
O1 O3

23.  Area of segment = x Area of parallelogram F13D
= x E2 x 2d
=
 So, the area between the first two divisions,
23
3
1 2 3

d
O  4O  O 

24.  Similarly the area between next two divisions,
and so on.
 Total area
24
 4O4  O5
3
3
2
Δ 
d
O
 4O2  2O3  4O4  ...... On 
3
1

d
O
3
5
3
4
2
n
 O  4O  O  ..... 2O  O ....
1

d
O
[
]

25.  Limitations :
 This rule is applicable only when the number of divisions must
be even.
 i.e. The number of ordinates must be odd.
25

26. Trapezoidal Rule Simpson’s Rule
The boundary between the
ordinate is considered to be
straight.
The boundary between the
ordinate is considered to be an
arc of a parabola.
There is no limitation. It can be
applied for any number of
ordinates.
This rule can be applied when
the number of ordinate must be
odd.
It gives an approximate result It gives a more accurate result
than the trapezoidal rule.
26

27.  The area of a closed traverse may be calculated from the
following methods :
1. The triangle method
2. The coordinates (X and Y)
3. The latitude and double meridian distance
4. The departure and total latitudes
27

28.  Triangle method :
 The polygon can be divided into simple figures, each of which
can be measured. The sum of the area of each figure will give
the total area.
28

29.  Triangle method :
 Using only triangles divides the area into fewer pieces and
simplifies calculations (i.e., fewer calculations are necessary).
29

30.  Triangle method :
 When all the sides are known, the area of the triangle is given
by
 Where,
and a,b,c are the lengths of the three sides BC, AC and AB.
30

31.  Triangle method :
 If the two sides and the included angle are known, the area is
given by
Area
or
or
31

32.  Triangle method :
 If the lengths of the base and perpendicular are known, the
area is given by
Area
32

33.  The coordinates (X and Y) :
 The area of a closed traverse may be calculated from the
independent coordinates (X and Y) method.
 The consecutive coordinates of a traverse are converted into
independent coordinates with reference to the coordinates of
the most westerly station.
 Thus, the whole traverse is transferred to the first quadrant.
33

34.  The coordinates (X and Y) :
 In fig. 1 the most westerly station.
34

35.  The coordinates (X and Y) :
 Then, the coordinates are arranged in determinant form as
follows :
35

36.  The coordinates (X and Y) :
 Double area = ∑A - ∑ B
 So, Required area = ½ ( ∑A - ∑ B )
 This way area of the closed traverse may be calculated.
36

37.  The latitude and double meridian distance (DMD):
 Refer the figure,
37
N

38.  The latitude and double meridian distance (DMD):
 Thus, the calculation of DMD :
1. DMD of first line = departure of first line
2. DMD of second line = DMD of first line + departure of first
line + departure of second line
3. DMD of any succeeding line = DMD of any preceding line +
departure of preceding line + departure of line itself
38

39.  The latitude and double meridian distance (DMD):
 Calculation of area :
1. Each DMD is multiplied by the latitude of that line.
2. The algebraic sum of these product is worked out.
3. This sum is equal to twice the area.
4. Half of this sum gives the required area of the traverse
ABCDE.
39

40.  The departure and total latitudes :
 The total latitude of a point is its distance from the reference
station measured parallel to the reference meridian.
 In figure, the reference station chosen in P through which the
reference meridian PN is passing.
40

41.  The departure and total latitudes :
 Total latitude of a point is equal to the algebraic sum of the
latitude of the preceding station and latitude of the preceding
line.
41

42.  The departure and total latitudes :
 Calculation of area :
1. The total latitude (the latitude with respect to the reference
point) of each station of the traverse is found out.
2. The algebraic sum of departures of the two lines meeting at a
station is determined.
3. The total latitude is multiplied by the algebraic sum of
departure, for each individual point.
42

43.  The departure and total latitudes :
 Calculation of area :
4. The algebraic sum of this product gives twice the area.
5. Half of this sum gives the required area of the traverse
PQRST.
 NOTE : The negative sign to the area has no significance.
43

44.  The planimeter is used for computation of area from a plotted
map.
44

45.  Parts of a planimeter :
45

46.  Tracing Arm :
 Tracing arm is an arm which manages the position of tracing
point at one end with the help of hinge.
 Tracing Point :
 Tracing point is the movable needle point which is connected
to tracing arm. This point is moved over the outline of area to
be measured.
46

47.  Anchor Arm :
 Anchor arm is used to manage the anchor position or needle
point position on the plan. Its one end is connected to weight
and needle point and other end to the integrating unit.
 Weight and Needle Point :
 It is also called as anchor. A fine needle point is located at the
base of heavy block. This needle point is anchored at required
station on the plan.
47

48.  Tangent Screw :
 Tangent screw is used to extend the tracing arm up to required
length.
 Index :
 Index is a location where all the measuring arrangements like
wheel, dial are located.
48

49.  Clamp :
 Clamp is used to fix the tracing arm in standard length without
any extension.
 Hinge :
 The tracing arm and anchor arm are connected by hinge to the
integrating unit. With the help of this hinge the arms can rotate
about their axes.
49

50.  Wheel :
 Wheel is fixed in the integrating unit which helps to measure
the tracing length. It is used to set zero on the scale.
 Dial :
 Dial is nothing but scale which is to be set zero at the initial
level using setting wheel.
50

51.  Vernier :
 Vernier is attached to the wheel as rounded drum with
graduations on it. It is divided into 100 parts.
51

52.  The vernier of the index mark is set to the exact graduation
marked on the tracer arm corresponding to the scale as given
in the table.
 Suppose the scale is 1:1, then the vernier of the index mark
should be set as per value given in the table against scale of
1:1.
52

53.  The anchor point is fixed firmly in the paper outside or inside
the figure. But it is always preferable to set the anchor point
outside the figure.
 If the area is very large, it can be divided into a number of
small divisions. It should be ensured that the tracing point is
easily able to reach every point on the boundary line.
53

54.  A good starting point is marked on the boundary line and the
tracing point is placed over it.
 Observe the counting disc, the wheel and the vernier and
record it as initial reading (IR).
 The tracing point is moved gently in a clockwise direction
along the boundary of the area till it returns to the starting
point.
54

55.  The number of times the zero mark of the dial passes the index
mark in a clockwise or anticlockwise direction should be
noticed.
 Finally, by observing the disc, wheel and vernier the final
reading (FR) is recorded.
 The difference of the final and initial reading gives the
required plan area A.
55

56.  A = M (FR - IR ± 10N + C)
 Where, M = Multiplier constant given in the table
(M = 100 cm2, natural scale 1 : 1)
 N = Number of times the zero mark of the dial passes the
index mark
 C = The additive constant given in the table
 FR=Final reading
 IR = Initial reading
56

57.  Note:
 N is considered +Ve when the zero of the dial passes the index
mark in a clockwise direction.
 N is considered -Ve when the zero of the dial passes the index
mark in an anticlockwise direction.
 The value of C is added only when the anchor point is inside
the figure.
57

58.  Note:
 Generally vernier is set as per scale 1:1 on the tracing arm and
the area of the land is obtained by multiplying the plan area by
the scale to which the plan is drawn.
 When anchor point is outside C = 0, if zero mark does not pass
the index mark, N = 0 and M = 100 (generally) then required
area A = 100 (FR - IR).
58

59.  The multiplier constant or the planimeter constant is equal to
the number of units of area per revolution of the roller. The
value of the constant for various settings of the tracing arm are
generally supplied by the instrument maker. When not
supplied, it can be calculated by the expression.
 M = 2mrL
 where M = the planimeter constant
 L = length of tracing arm & r =the radius of the roller or wheel
59

60.  The value of M is generally kept at 100 cm2.
 For any other setting of the tracing arm, the value of M can be
determined by traversing the perimeter of a figure of known
area (A), with anchor point outside the figure.
 Then,
 M
60

61.  M
 Where n' = change in the wheel readings.
 It is to be noted that the value of M and C depends upon the
length L which is adjustable.
 The manufactures always supply the values of the vernier
setting on the tracing arm with corresponding values of M and
C.
61

62.  The following table is an extract from the values for a typical
planimeter.
62
Scale Vernier position
on tracer bar
Area for one revolution of
measuring wheel (M)
Constant (C)
Scale Actual
1:1 33.33 100 cm2 100 cm2 23.254
1:200 33.33 0.4 m2 - 23.254
1:400 20.83 1 m2 - 27.133
1:500 26.67 2 m2 - 24.637
1:1000 33.33 10 m2 - 23.547

63.  When the tracing point is moved along a circle without
rotation of the wheel (i.e. when the wheel slides without any
change in reading), the circle is known as the 'zero circle' or
'circle of correction’
63

64.  The zero circle is obtained by moving the tracing point in such
a way that the tracing arm makes an angle of 90° with the
anchor arm.
64

65.  The anchor point A is the centre of rotation and AT (R) is
known as the radius of the zero circle.
 When the anchor point is inside the figure, the area of the
zero-circle is added to the area computed by planimeter.
 Area of zero circle,
 A = M × C
 Where, M =multiplier value given in table
 C=constant given in table
65

66.  Computations of volumes for all types of projects for their
designing and estimation of earthwork, is commonly required
during construction work such as roads, railways, canals etc.
66

67.  For estimation of the volume of earthwork, cross-sections are
taken at right angles to a fixed line (generally, the centre line)
which runs longitudinally through the earthwork.
67

68.  Level section :
68

69.  Level section :
 In this case the ground is level transversely.
69

70.  Two-level section :
70
W2 W1

71.  Two-level section :
 In this case, the ground is sloping transversely, but the slope of
the ground does not intersect the formation level.
71

72.  Two-level section :
72

73.  Three-level section :
73
W2 W1

74.  Three-level section :
74

75.  Side hill two-level section :
 In this case the ground is sloping transversely, but the slope of
the ground intersects the formation level such that one portion
of the area is in cutting and the other in filling (part cut and
part fill).
75

76.  Side hill two-level section :
76

77.  Side hill two-level section :
77

78.  Side hill two-level section :
78

79.  Multi-level section :
 In this case, the transverse slope of the ground is not uniform
but-has multiple cross-slopes as is clear from the figure.
79

80.  Multi-level section :
 The notes regarding the cross-section are recorded as follows:
80

81.  Multi-level section :
 The numerator denotes cutting (+ve) or filling (-ve) at the
various points, and the denominator their horizontal distances
from the centre line of the section.
 The area of the section is calculated from these notes by co-
ordinate method. The co-ordinates may be written in the
determinant form irrespective of the signs.
81

82.  Multi-level section :
 Let Σ F= sum of the product of the co-ordinates joined by full
lines.
 Σ D= sum of the products of the co-ordinates joined by dotted
lines.
 Then, A= 1/2 (ΣF- ΣD)
82

83.  In many civil engineering projects, earthwork involves the
excavation, removal and dumping of earth, therefore it is
required to make good estimates of volumes of earthwork.
 After calculation of cross-sectional areas, the volume of
earthwork is calculated by :
1. Trapezoidal formula
2. Prismoidal formula
83

84.  The number of cross-sections giving the areas may be odd or
even. Since the areas at ends are averaged in this formula,
therefore, it is also known as Average End Area formula.
84
A1 A2 An-1 An

85.  In order to apply the prismoidal formula, it is necessary to
have odd number of sections giving the areas.
 If there be even areas, the prismoidal formula may be applied
to odd number of areas and the volume between the last two
sections may be obtained separately by trapezoidal formula
and added.
85

86.  When the more exact volume is required to be computed, the
prismoidal formula given below , may be used,
86

87.  The volume obtained by the prismoidal formula is more
accurate than has obtained by the trapezoidal rule. But, the
prismoidal formula is applicable when there are an odd
number of sections.
87

88.  However, it is more inconvenient to use the prismoidal
formula than the trapezoidal formula. Instead of calculating
the volume by the prismoidal formula, it is generally more
convenient to calculate the volume by the trapezoidal formula
and apply a correction called the prismoidal correction.
 The prismoidal correction is equal to the difference between
the volumes as calculated by the end-area formula and the
prismoidal formula.
88

89.  As the volume obtained by the trapezoidal formula is always
greater than that obtained by the prismoidal formula, the
prismoidal correction is always negative.
89

90.  Cp = volume by end area formula - volume by prismoidal
formula
 Cp (A1 +4 Am + A2)
 Cp (3A1 +3A2-A1-4Am -A2)
OR
 Cp (A1 -2Am +A2)
 Where Am = area of mid section
90
(A1 + A2)

91.  Level Section :
 Two–Level Section :
91

92.  Three–Level Section :
92

93.  Side-Hill Two–Level Section :
93

94.  The average end-area method for calculating volumes is based
on the assumptions that the sections are parallel to each other
and normal to the centre-line.
 But, on curves, the centre-line is curved and the cross-sections
are set out in the radial direction and therefore, they are not
parallel to one another. Hence, the volumes computed by the
usual methods, assuming the end faces parallel require a
correction for the curvature of the central line.
94

95.  The calculation of the volumes along the curved line, is made
by the Pappu's theorem.
 According to Pappu's theorem, the volume swept out by an
area evolving about an axis is equal to the product of the area
and the length of the path traced by the centroid of the area.
 Pappu's theorem can be used to compute the volume of earth
in cutting or filling on curves.
95

96.  Volume = Area x distance traced by the centroid
Let, e1 = distance of centroid of area A1 from the centre line
e2 =distance of centroid of area A2 from the centre line
= mean distance of the area A1 and A2 from the centre
line
D=distance between two curved centre-lines
R = radius of the curved central line
96

97.  The length of the path traced by the centroid may be
calculated as,
 D
 The volume of curve is calculated as,
97

98.  Level Section :
 No correction is required, since the area is symmetrical about
the centre-line, and therefore,
 Cc=0
98

99.  Two level section and Three level section :
99

100.  Side–Hill Two–Level Section :
100

101.  The plane containing any contour, represents a horizontal
plane and the area bounded by a contour is treated as the area
of the cross-section. The contour interval is the vertical
distance between any two adjacent cross-sections bounded by
the contours. The area bounded by the contour is measured by
a planimeter.
101

102.  Reservoirs are made for water supply, irrigation, hydropower
etc. A contour map is very useful to study the possible location
of a reservoir and the volume of water to be confined. All the
contours are closed lines within the reservoir area.
102

103.  The area A1 , A2, A3, ..., An between successive contour lines
can be determined by a planimeter and if h is the contour
interval, the capacity of the reservoir can be estimated either
by the prismoidal formula or by the trapezoidal formula.
 In practice, the capacity of a reservoir is measured in terms of
volume of water stored up to full reservoir level (F.R.L.)
which is the level of water at its full capacity.
103

104.  The volume of borrow pits and excavations for large tanks,
basements, etc. with vertical sides, can be determined from the
spot levels of the ground before starting the excavation and the
required formation level.
 The volume of a borrow pit or of the excavation for the
foundation of a building, is frequently found by dividing the
surface area into triangles, squares or rectangles.
104

105.  The points where different lines intersect one another are
called corner points. These corner points are marked on the
ground, spot levels are taken at all the corner points with a
levelling instrument.
 The depth of excavation or height of filling at every corner
point may be determined from the ground level and the
required formation level.
105

106.  This method of estimating the earthwork quantities is also
known as the unit area method.
 Refer the figure,
106
p1
s1
q1
r1
p
s
q
r

107.  In this method the area is divided into regular figures such as
squares, rectangles or triangles and the levels of corners of the
figures are measured before and after the construction.
 Thus the cut or fill at each corner is a known parameter. The
corners of the figures may be at different elevation but lie in
the same plane. The surface of the ground within the figure,
therefore lies in an inclined plane.
107

108.  If it is desired to grade down to a level surface at a certain
distance below the lowest corner of the figure, say pqrs.
 Refer the figure,
108
m1
n1 o1
m
n o

109.  The earth to be moved will be a right truncated prism. If the
depth of the cuts are p1, q1, r1, and s1 respectively, and A is the
area of the figure pqrs then the volume is given by,
 V = base area x mean height
 V = A x p1 + q1 + r1 + s1 cu.m.
109
4

110.  Similarly, volume of the figure mno is,
 V = A x m1 + n1 + o1 cu.m.
110
3

111. 11
1