2. Introduction
The computation of area is very essential to determine the
catchment area of river, dam and reservoir. Its also important for
planning and management of any engineering project.
For initial reports and estimates, low precision methods can be
used.
When a high level of accuracy is required, a professional
engineer or a land surveyor should be employed.
The area is expressed in ft2, m2, km2, acres, hectares.
3. Methods to Compute Area
The method of computation of area depends on the shape of the
boundary of the surveyed area and accuracy required.
If the plan is bounded by straight boundaries, it can be tackle by
subdividing the total area into simple geometrical shapes, like
triangle, rectangle, trapezoidal etc… and the area of the figure
are computed from the dimensions.
If the boundaries are irregular, they are replaced by short straight
boundaries and the area is computed using approximate method.
While if the boundaries are very irregular, the area can be
determined by using planimeter.
4. Computation of Area
The surveyed area may be calculated from plotted plan by
following rules.
1. Mid ordinate rule
2. Average ordinate rule
3. Trapezoidal rule
4. Simpson’s one third rule
5. Mid Ordinate Rule
The method is used with the assumption that the boundaries
between the
edge of the ordinates are straight lines.
The base line is divided into a number of divisions and the
ordinates are
measured at the mid points of each division.
6. The area is calculated from following formula,
Area = ∆ = Common distance x Sum of mid ordinates
= (h1 x d) + (h2 x d) + …… + (hn x d)
= d (h1+ h2+….. +hn)
Where,
n = Number of divisions
d = common distance between ordinates
h1, h2, … hn = Mid ordinates
7. Average Ordinate Rule
This rule also assume that the boundaries between the edges of
the ordinates are straight lines. The offsets are measured to each
of the points of the divisions of the base line.
The area is given by following equation,
Area = ∆ = average ordinate x Length of the base
8. Trapezoidal Rule
This rule is based on the assumption that the figures are
trapezoids. The rule is more accurate than the previous two rules
which are approximate versions of the trapezoidal rule.
9. The area of the first trapezoid is given by
Similarly, the area of the second trapezoid is given by
So, the total area isgivenby
∆ = ∆1 + ∆2 + …. ∆n
Total area = (O1 + 2O2 + 2O3 + 2O4 +… + 2On-1 + On) x (d/2)
= (O1 + On + 2(O2 + O3 + O4 +…+ On-1)) x (d/2)
= (Common distance/2) x [(1st ordinate +last ordinate) + 2(sum
of other ordinates)]
10. Simpson’s One Third Rule
This rule assumes that the short lengths of boundary between
the ordinates are parabolic arcs. So this rule is some times
called the parabolic rule.
This method is more useful when the boundary line departs
considerably from straight line.
11. Here, O1, O2, O3 = Three consecutiveordinates
d = Common distance between the ordinates
Now, Area of AF2DC = Area of AFDC + Area of segment F2DEf
Area of trapezium =
Area of segment =
12. So, the area between the first two divisions,
Similarly, the area between next two divisions,
Total area
= (Common distance/3) x [(1st ordinate + last ordinate) + 4(sum
of even ordinates) + 2(sum of odd ordinates)]
13. Computation of Volume
The volume of earth work is calculated by following two method
after calculation of cross sectional area,
1. Trapezoidal rule
2. Prismoidal rule
14. Trapezoidal rule
Volume,
V= (d/2) x [A1+An+ 2(A2+A3+…..+An-1)]
= (Common distance/2) x [(1st section area + last section area) +
2(sum of area of other section)]
d
15. Prismoidal rule
Volume,
V = (d/3) x [A1+An+ 4(A2+A4+…..+An-1) + 2 (A3+A5+…..+An-2)]
Limitation:
The prismoidal formula is applicable when there are odd number of
sections. If the number of sections are even, the section is treated
separately and area is calculated according to the trapezoidal rule.
