GEOMATICS ENGINEERING

CHAPTER 6

Methods of Areas and Volume Calculation

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Areas Calculation

• The term ‘area’ in the context of surveying refers to the

area of a tract of land projection upon the horizontal

plane, and not to the actual area of the land surface.

• Area may be expressed in the following units.

• Square meters

• Hectares (1 hectare = 10,000 m2)

• Square feet

• Acres (1 acre = 4840 sq-yard)

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Computation of Area from Plotted Plan

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• A large square or rectangle is formed within the area in

the plan. Then coordinates are drawn at regular intervals

from the side of the square to the curved boundary. The

middle area is calculated in the usual way. The boundary

area is calculated according to one of the following rules:

1. The mid-ordinate rule

2. The average ordinate rule

3. The trapezoidal rule

4. Simpson’s rule

The Mid-Ordinate Rule

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The Average Ordinate Method

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The Trapezoidal Rule

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Simpson’s Rule • In this rule, the boundaries between the ends of ordinates are

assumed to form an arc of a parabolic. Hence Simpson’s rule is sometimes called the parabolic rule.

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Simpson’s Rule

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Comparison Between Trapezoidal and Simpson’s Rule

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Example 1 • The following offsets were taken from a chain line to an irregular boundary

line at an interval of 10 m.

• 0, 2.50, 3.50, 5.00, 4.60, 3.20, 0.0

• Compute the area between the chain line, the irregular boundary line and the

end offsets by:

1. The mid-ordinate rule

2. The average ordinate rule

3. The trapezoidal rule

4. Simpson’s rule

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Solution to Example 1: By mid-ordinate rule

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Solution to Example 1: By average-ordinate and Trapezoidal rule

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Solution to Example 1: By Simpson’s rule

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Example 2 • The following offsets were taken at 15 m intervals from a

survey line to an irregular boundary line:

• 3.50, 4.30, 6.75, 5.25, 7.50, 8.80, 7.90, 6.40, 4.40, 3.25 m

• Calculate the area by

• The trapezoidal rule

• Simpson’s rule

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Solution to Example 2: by trapezoidal rule

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Solution to Example 2: by Simpson’s rule

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COMPUTATION OF VOLUME

Introduction • Areas and Volumes are often required in the context of

design, e.g. we might need the surface area of a lake, the

area of crops, of a car park or a roof, the volume of a dam

embankment, or of a road cutting.

• Volumes are often calculated by integrating the area at

regular intervals e.g. along a road centerline, or by using

regularly spaced contours. We simply use what you

already know about numerical integration from numerical

methods).

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Introduction • For computation of the volume of earth work, the sectional areas of the

cross-section which are taken transverse to the longitudinal section during profile leveling are first calculated.

• Cross-sections may be to different types, i. Level

ii. Two-level

iii. Three level

iv. Side hill two level, and

v. Multi-level.

• After calculation of cross-sectional areas, the volume of earth work is calculated.

i. The trapezoidal (or average end area)

ii. The prismoidal rule

• The prismoidal rule gives the correct volume directly.

• The trapezoidal rule does not give the correct volume. Prismoidal correction should be applied for this purpose. This correction is always subtractive.

• Cutting is denoted by a positive sign and filling by a negative sign

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Formulae for Calculation of Cross-Sectional Area

• Level Section

• When the ground is level along the transverse direction.

• Man-made structures usually have constant side slopes.

• Example: calculate the sectional area of an embankment 10 m wide, a

side slope of 2:1. the ground is level in a transverse direction to the center

line. The central height of the embankment is 2.5 m.

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Example Solution:

b = 10 m, s = 2, h = 2.5 m

Cross-sectional area = (10+2*2.5)*2.5

= 37.5 m2

• Two-level section • In general, ground is not flat. If the cross-fall is constant

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• Two-level section

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• Three-level Section • Non-Constant Cross-fall (e.g. centerline plus one point on either side.

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• Partially in Cut and Fill

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A2

Area in Cutting

Area in Filling

• Partially Cut and Fill

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• Many Level section • Other, more complex cases are possible, e.g. n-Level Section. However,

these are handled more conveniently using the Coordinate Method

(described under "Areas"), and a computer. (not covered here)

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Formula for Calculation of Volume

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Formula for Calculation of Volume

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Example 1 for volume calculation

• An embankment of width 10 m and side slopes 1.5: 1 is

required to be made on a ground which is level in a

direction transverse to the center line. The central heights

at 40 m intervals are as follows:

• 0.90, 1.25, 2.15, 2.50, 1.85, 1.35, and 0.85

• Calculate the volume of earth work according to

• Trapezoidal formula, and

• The prismoidal formula

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Example 2 for volume calculation

• An excavation is to be made for a reservoir 40 m long and

30 m wide at the bottom. The side slope of the excavation

has to be 2:1. Calculate the volume of earthwork if the

depth of excavation is 5 m. Assume level ground at the

site.

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Example 2 for volume calculation

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How about trapezoidal formula?

Example 3 for volume calculation

• The areas enclosed by the contours in a lake are as

follows:

• Calculate the volume of water between the contours 270

m and 290 m by:

• The trapezoidal formula

• Prismoidal formula

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Contour (m) 270 275 280 285 290

Area (m2) 2050 8400 16300 24600 31500

Example 3 for volume calculation

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