Measurement of Area and Volume (Mid Ordinate Rule, Average Offset Rule, Trapezoidal Rule, Simpson’s Rule)

  Introduction  

Many surveys have the primary goal of finding the area of the surveyed tract and the amount of earthwork.

  Measurement of Area  

  • The method adopted for measurement of area and volume depends upon
    1. Accuracy required
    2. Shape/ Geometry of track.
  • Measurement can be done by either
    1. By field measurement
    2. By plan measurement
  • If the plan is enclosed by a straight line, it can be divided into geometrical figures, like: triangle, rectangle, square etc. The area of these figures can be determined by using standard formulas.
  • But if the boundaries are irregular, then approximate methods are being used.

Note: Planimeter meter is used to determine areas of irregular shape.

Computation of Area of Geometrical Figures

1.Rectangle

Area= length(L) × width(b)

2.Square

Area= length(L) × width(b)

3.Triangle

Area= half of the base(b) × perpendicular height(h)

4.Circle

Area = Π × (Radius)2

5.Parallelogram

Area= base × perpendicular height

6.Trapezoid

Area= half of the sum of parallel sides ×  perpendicular height

Measurement of Area and Volume,Mid Ordinate Rule, Average Offset Rule, Trapezoidal Rule/End Area Method, Simpson's Rule, Prismoidal Rule

7.Trapezium

Area (ABCD) = Area(ABC) + Area(ACD)

Measurement of Area and Volume,Mid Ordinate Rule, Average Offset Rule, Trapezoidal Rule/End Area Method, Simpson's Rule, Prismoidal Rule

8.Oblique Triangle

\(A=\sqrt{s(s-a)(s-b)(s-c)}\)

where a, b, care the sides.

\(s=\frac{a+b+c}{2}\)

Measurement of Area and Volume,Mid Ordinate Rule, Average Offset Rule, Trapezoidal Rule/End Area Method, Simpson's Rule, Prismoidal Rule

9.Area of a Segment

Area of segment (ACB) = Area of sector (ACBO) – Area of triangle (AOB)

\(A=\frac{\Pi R^{2}}{360^{\circ}}\times \alpha -\frac{1}{2}\times 2Rsin\frac{\alpha }{2}\times Rcos\frac{\alpha }{2}\)

\(A=R^{2}\left ( \frac{\Pi \alpha }{360^{\circ}} -\frac{sin}{2}\right )\)

\(A=R^{2}\left ( \frac{\Pi \alpha }{360^{\circ}} -\frac{sin}{2}\right )\)

Area of segment min

 

10.Regular Polygon

Area= length of perimeter ×  half of the perpendicular distance from the centre of sides.

\(A=\frac{nL^{2}}{4}cot^{2}\frac{180^{\circ}}{n}\)

\(A=\frac{nL^{2}}{4}cot^{2}\frac{180^{\circ}}{n}\)

where n is the number of sides and L the length of one sides

Regular Polygon min


Computation of Area of Irregular Shape

1.Mid Ordinate Rule

If offsets h1, h2, …., hn-1. are measured at the mid point of each division.

Area = average ordinate × length of base.

Mid Ordinate Rule Average Offset Method Trapezoidal Rulee

\(A=\frac{h_1+h_2+….+h_{n-1}}{n-1}L\).

∴L=(n-1)d

\(A=\frac{h_1+h_2+….+h_{n-1}}{n-1}(n-1)d\)

\(A=(h_1+h_2+…+h_{n-1})d\)

where

n-1 = number of division

d = distance between the two perpendicular offsets.

L = length of base line {L=(n-1)d}

2.Average Offset Rule

If offsets O1, O2, …., On. are measured at the mid point of each division and spaced apart at equal distance d.

Mid Ordinate Rule Average Offset Method Trapezoidal Rulee

Area = average of all ordinate × length of base.

\(A=\frac{O_1+O_2+….+O_n}{n}L\)

\(A=\frac{O_1+O_2+….+O_n}{n}(n-1)d\)

\(A=\frac{(n-1)d}{n}\Sigma O_i\)

3.Trapezoidal Rule

The accuracy of this method is more than mid-ordinate and average ordinate method.

Area of first trapezoid \(=\frac{O_1+O_2}{2}d\).

Area of second trapezoid \(=\frac{O_2+O_3}{2}d\).

Area of last trapezoid \(=\frac{O_{n-1}+O_n}{2}d\).

Total area of trapezoid

\(A=\frac{1}{2}(O_1+O_2)d+\frac{1}{2}(O_2+O_3)d+….+\frac{1}{2}(O_{n-1}+O_n)d\)

\(A=\frac{1}{2}d[O_1+2O_2+2O_3+2O_{n-1}+O_n]\)

\(A=d[\frac{O_1+O_N}{2}+O_2+O_3+…O_{n-1}]\)

4.Simpson’s One Third Rule

  • If the boundaries of irregular tract are curved then Simpson method is preferred over the trapezoidal rule to calculate the area of given track.
  • In this the curved boundary is considered to be a parabolic arch.

Simpson's One Third Rule

\(A=\frac{d}{3}[(O_1+O_n)+4(O_2+O_4+…+O_{n-1})+2(O_3+O_5+…+O_{n-2})]\)

Note

  • To apply Simpson’s rule offsets should be odd in number.
  • To work with even no of offsets Simpson’s should be apply upto last offsets and remaining area should be calculated using trapezoidal rule.

  Measurement Of Volume 

The computation of the volume of different quantities is required for planning & designing  of various engineering work .

  1. Earthwork for highway, railway, retaining walls.
  2. Volume for reservoir capacity
  3. Concreting work
  4. Storage requirement

Computation of Volume of Irregular Shape

1.Trapezoidal Rule / End Area Method

This method is also called as end are method.

\(V=d[\frac{A_1+A_n}{2}+A_2+A_3+…A_{n-1}]\)

2.Prismoidal / Simpson’s  Rule

\(V=\frac{d}{3}[(A_1+A_n)+4(A_2+A_4+…+A_{n-1})+2(A_3+A_5+…+A_{n-2})]\)

Note

  • This formula usually gives less volume than trapezoidal formula.
  • This formula is not suitable for rock excavation and concrete work.
  • To apply Simpson’s rule offsets should be odd in number.
  • To work with even no of offsets Simpson’s should be apply upto last offsets and remaining volume should be calculated using trapezoidal rule.

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