Methods of computing area

Content : Introduction Methods of computing area Average ordinate rule Mid ordinate rule simpson’s rule

Introduction In civil engineering work such as design of bridges , dam ,reservoirs etc. The area of catchment of a river is required. For road and railways land is to be acquired on the basis of area. Thus , finding areas is essential part of surveying. It may be noted that the area to be found is the projected area upon the horizontal plane.

units used for finding the area are square , meters , hectare , acres etc. 100 Sq. m=1 are 100 are=1 hectare =10000 Sq. m 1 acre = 4047 Sq. m = 2.5 vigha 1 vigha =16 guntha 1 acre = 40 guntha 1 Hectare =2.471 acres 1 Sq. m=10000000 Sq. m

Computation of area from plotted plan Boundary area can be calculated as one of the following rule: The mid-ordinate rule The average ordinate rule The trapezoidal rule Simpson’s rule

Methods of computing area Computation of area by taking offsets Mid-ordinate rule Average ordinate rule Trapezoidal rule Simpson’s rule Computation of area by planimeter Computation of area by zero circles

Computation of area by taking offsets: Various methods of computation of area by taking offsets are Mid-ordinate rule Average ordinate rule Trapezoidal rule Simpson’s rule

Mid-ordinate rule In this method the base line is divided into a number of divisions and the ordinates are measured at the points of each divisions . Boundaries between the offsets are considered straight lines.

Where h1,h2,h3,…………=mid ordinates d=distance of each division L=length of base line= nd n=number of division

Average ordinate rule This rule also assumes that the boundaries between the extremities of the ordinates are straight lines.

Where h0,h1,h2,……=ordinates of offsets d=distance of each division n=number of division n+1=number of offsets L=length of base line=nd

Trapezoidal rule In this method , entire area is divided in to trapezoids . The rule is more accurate than the previous two rules.

which is known as trapezoidal rule.

Example: series of offsets were taken from a chain line to an boundary , interval of 15 m , in the following order. 0,1.65,3.50,2.70,4.65,3.60,3.95,4.85m Compute the area by trapezoidal rule. Solution:

Simpson’s rule This rule assumes that the short lengths of boundary between the ordinates are parabolic arcs.

For simpson’s rule , the number of ordinate must be odd. simpson’s rule is:

Application: Simpson’s rule used for find the earthwork volume using contour maps.it gives more accurate area. Trapezoidal rule can be applied for any number of ordinates. It gives an approximate area A planimeter is used to measure the area of any shape with more accuracy. Zero circle is used when the tracing point is moved , no rotation of wheel will take place .

Example : Following perpendicular offsets were taken from a chain line a curved boundary line at an interval of 10 m. 0,7.26,5.83,6.45,7.20,8.18,8.0,0 compute the area by simpsons rule Solution: To find area by simpson’s rule , number of offsets must be odd. Here we have 8 offsets. Therefore , for offsets h0 to h6 apply simpson’s rule and for offsets h6 and h7 apply trapezoidal rule.

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