Computation of Area and volume

Computation of Area and volume

• 2.
  Introduction  The computationof 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 ComputeArea  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
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  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.
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  The area iscalculated 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
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  Average Ordinate Rule Thisrule 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
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  Trapezoidal Rule This ruleis 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.
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  The area ofthe 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 ThirdRule  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.
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   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 =
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  So, the areabetween 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
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  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.
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