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Thursday, March 21, 2019

Computation of area in surveying


COMPUTATION OF AREA


INTRODUCTION:

The term ‘area’ in surveying refers to the area of a tract of land projected upon the horizontal plane, and not to the actual area of the land surface.
The following are some unit in which area in expressed – Square Meters (Sqm.), Hectares (1 hectare = 10,000 Sqm.), Square Feet, Acres (1 acre = 43.560 Sq. ft.)
METHODS OF COMPUTATION OF AREA:
Generally, area can be calculated from the from the following method –
§  COMPUTATION AREA FROM FIELD NOTES:


­  Step – 01

Ø  During the cross-stuff survey, the area of the field can be directly calculated from field notes.
Ø  During the survey wok the whole area is divided into some geometrical figures, such as squares, rectangles, triangles, and trapeziums, and then the area is calculating by using the following formulas –


a)      Area of Squares – Area = a2
Where ‘a’ is the Sides of the Square

a)      Area of Triangle – Area = ‘ax b’ or ‘w x l’
    
‘Or’ Area of Triangle = 
                                                    


Where, a, b and c are the sides of triangle
And s = (a +b + c)/2

a)      Area of Trapezium = ½ x (a + b) x h’ 
Where a and b are the parallel sides and h is the perpendicular distance between them.
­  Step – 02
Ø  The Area along the boundaries is calculated as follows –
In figure,
O1, O2 = Ordinates
X1, X2  =  Chainages
Then the area of the shaded portion = 



In the same way, the area between all pairs of ordinates are calculated and added to obtain the total boundary area.

Therefore, the Total Area of the field =
                               Area of geometrical figure (ABCD) + Boundary area (ABFEA)
                                                    (Step – 01)                     +             (Step – 02)                 


·        Example
A page of the field book of a cross-staff survey is given as follows. Plot the required figure and calculate the relevant area.

 

·        Solution


The figure can be plotted as follows –
The area is calculated as follows –

·         Area of the Field = 6455 sq.m

§  COMPUTATION OF AREA FROM PLOTTED PLAN:

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 geometrical shape and the area is calculated as follows –
1)         By Dividing the Area into Triangles –

·      The triangles are drawn so as to equalize the irregular boundary line (as shown in figure).
·      Then the base and altitude of the triangles are determined according to the scale to which the plane was drawn.
·      The area of the triangles are calculated from area = ½ x base x altitude.
·      The areas of the no of divided triangles added to obtain the total area.

1)            By Dividing the Area into Squares–

·      By this method squares of equal are lined out on a piece of tracing paper.
·      Each of the squares represents a unit area which could be 1 Sq. cm or 1 Sq. m.
·      Then the tracing paper is placed over the plan and the number of full squares is counted.
·      The total area is then calculated by multiplying the number of squares by their unit area of each squares.
1)             By Drawing Parallel Lines and Converting them to Rectangles –

·      By this method, a series of equidistant parallel lines are drawn on a tracing paper.
·      The constant distance represents a meter or centimetre.
·      The tracing paper is placed over the plan in such a way that the area is enclosed between the two parallel lines at the top and bottom.
·      Thus, the area is divided into a number of stripes.
·      The curved ends of the stripes are replaced by perpendicular lines by give and take principles and a number of rectangles are formed.
·      The sum of the lengths of the rectangles is then calculated –
The required area = Length of Rectangles x Constant Distance

­  Case 02
 1. 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. The boundary area is calculated according to one of the following rules –


2.       The Average Ordinate Rule
3.       The Trapezoidal Rule
4.       Simpson’s Rule

The Mid-Ordinate Rule – 
In the above figure –
·               O1 , O2 , O3  ........ On  =  Are Ordinates at equal intervals.
·                                                                       l    = length of Base Line.
·                                                                      d    = Common Distance between ordinates.
·               h1 , h2 , h3,  ........, hn   =  Are Mid-Ordinates.

Then the Area of the Plot = h1 x d + h2 x d + ........ + hn x d
 = d (h1 + h2 + ........ + hn)
AREA   = Common Distance x Sum of mid-ordinates
1.       The Average Ordinate Rule
In the above figure –
·                O1 , O2 , O3  ........ On  = Are Ordinates or Offsets at regular intervals.
·                                                                       l    = length of Base Line.
·                                                                       n   = Number of divisions
      (n+l)  = Number of ordinates.
·              h1 , h2 , h3 , ........ hn    =  Are Mid-Ordinates.

Then the Area of the Plot =     
1.       The Trapezoidal Rule –
During the application of The Trapezoidal Rule boundaries between the ends of the ordinates are assumed to be straight. Thus the areas enclosed between the base line and the irregular boundary line are considered as trapezoids.
In the above figure –
·      O1, O2  ........ On  = Are Ordinates or Offsets at regular intervals.                    d    = Common distance
Then the Area of the Plot =     
Thus the Trapezoidal Rule may be started as follows –
·         To the sum of the first and the last ordinate, twice the sum of the intermediate ordinates is added.
·         This total sum is multiplied by the common distance.
·         Half of this product is the required area.
There are no such limitations for this rule and can be applied for any number of ordinates.
1.       Simpson’s Rule –
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.
In the above figure –
·   O1, O2, O3  = Are Ordinates or Offsets at regular intervals.
·   d    = Common distance between the ordinates
Area AFeDC  = Area of trapezium AFDC  + Area of segment FeDEF


·          Thus, the rule stated as follows –
Ø  The sum of the first and the last ordinate, four times the sum of the even ordinates and twice the sum of the remaining odd ordinates are added.
Ø  This total sum is then multiplied by the common distance.
Ø  One third of this product is the required area.
This rule is applicable only when the number division is even i.e. the number of ordinates is odd.
­  Example
The following offsets were taken from a chain line to an irregular boundary line at an interval of 10 m –
                                   0, 3.50, 4.50, 6.00, 4.20, 2.90, 0 m
Compute the area between the chain line the irregular boundary line and the end off set by –
A.     The Mid-ordinate Rule
B.      The Average-Ordinate Rule
C.      The Trapezoidal Rule
D.     Simpson’s Rule
­  Solution
A.  By Mid-Ordinate Rule –
The mid-ordinates are –
A.  By Average-Ordinate Rule –
Here, d = 10 m & n = 6 nos (No of divisions)
Base Length = 10 X 6 = 60 m
Number of Ordinates = 7
A.  By Trapezoidal Rule –
Here, d = 10 m
A.   By Simpson’s Rule –
 Here, d = 10 m





























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