Visit for more Learning Resources Advance Surveying Methods of computing area Prepared by : MR. S.N. NANAWARE Visit for more Learning Resources

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

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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Planimeter The area of the land in plane surveying is determined by various methods. This area is determined by a particular method is used for many purposes such as to determine the quantities of earthwork , fixing the boundary of the field and thus calculating the area based upon which the revenue is paid . The metric units of area are Sq. mt. or hectares.

(continue) A planimeter is a precise instrument which measures the area of plan of any shape with more accuracy. There are two types of planimeter: Amsler polar planimeter Roller planimeter The most commonly used type is the polar planimeter.

The various component parts of a polar planimeter is as follows: Tracing arrow and Tracing point Anchor arm Anchor point Rotating wheel Graduated drum Disc Magnifier

construction: It is consists of two arms hinged at a point known as the pivot point. One of the two arm is anchor arm , whose length is generally fixed . Anchor arm has got anchor point which is fixed either inside the plan or outside the plan depending upon tracing arm reach. . Other arm is called tracing arm having tracing point at its end. The length of the tracing arm is varied by means of a fixed screw which is attached with a slow motion screw.

procedure : To find the area of the plan , the anchor point is either placed outside or inside the area depending upon whether the area is small or big. A point is then marked on the boundary of area and the tracing point is kept over it. The initial reading of the wheel is then taken . Then the tracing point is moved along the boundary till it comes to the original point. Then the final reading is noted.

(continue) Where F=final reading I=initial reading M= A multiplying constant. It is equal to area per revolution of the roller. N= The number of times the zero marks of dial passed the fixed index mark. C=instrument constant

Example : An area was measured using a planimeter with anchor point inside the figure . The initial and final readings were 4.129 and 6.387 . The zero mark on disc passed the fixed index mark once in clockwise direction . If the value of M= 100 Sq. cm and constant of the instrument =24.686.calculate the area of the figure. Solution: Here, I=4.129 F=6.387 M=100 Final reading Initial reading M N Constant 6.387 4.129 100 sq. cm 24.686

(Continue) C= 24.686

Zero circle It is also known as circle of correction. It is defined as the circle round the circumferences of which if the tracing point is moved , no rotation of the wheel will take place , but the wheel will slide on the paper without changing the reading. The anchor point is the centre of the zero circle and the line joining the anchor point is its radius. The area of zero circle may be determined by following methods

(continue) By using formula: (a)Area of zero circle= M C Where M= multiplier C = the constant (b)Area of zero circle= Where L=the length of tracing arm L1=the distance between the hinge and the wheel R=the length of anchor arm

Example : Find the area of zero circle from the following taken using planimeter having M=100 Sq. cm observations: Position of anchor point Initial reading Final reading Outside the figure 8.596 4.149 Inside the figure 3.944 6.534

(continue) Solution: Here , M=100 , I=8. 596 , F = 4 (continue) Solution: Here , M=100 , I=8.596 , F = 4.149 ,C=0 When the anchor point is inside, I=3.944 , F=6.534 , N=-2

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Reference : R.B.Khasiya Mahajan publication R.Subramanian Oxford university press R.P.Rethliya Atul prakashan

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