Methods of computing area q Content Introduction Methods

Methods of computing area

q. 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.

Ø 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 Ø Computation of area by taking offsets 1. Mid-ordinate rule 2. Average ordinate rule 3. Trapezoidal rule 4. Simpson’s rule Ø Computation of area by planimeter Ø Computation of area by zero circles

Ø Computation of area by taking offsets: Various methods of computation of area by taking offsets are I. Mid-ordinate rule II. Average ordinate rule III. Trapezoidal rule IV. 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 h 1, h 2, h 3, …………=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 h 0, h 1, h 2, ……=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.

Ø 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. 85 m 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:

q. 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 h 0 to h 6 apply simpson’s rule and for offsets h 6 and h 7 apply trapezoidal rule.

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