Methods of Calculation of Areas in Surveying | Simpson’s Rule



Calculation of Areas in Surveying | Simpson’s Rule

In one of my previous articles, I discussed Midpoint Ordinate Rule and Average Ordinate Rule in detail with an example and listed out various important methods used for the calculation of areas in Surveying. In this article, we will deal with the next important method (rule) i.e. Simpson’s Rule along with a numerical example used for the calculation of areas in the field of Surveying.

Here are the five important rules (Methods) used for the calculation of areas in Surveying:

  1. Midpoint ordinate rule
  2. Average ordinate rule
  3. Simpson’s rule
  4. Trapezoidal rule
  5. Graphical rule

Simpson’s Rule

Statement

It states that, sum of first and last ordinates has to be done. Add twice the sum of remaining odd ordinates and four times the sum of remaining even ordinates. Multiply to this total sum by 1/3rd of the common distance between the ordinates which gives the required area.

Where O1, O2, O3, …. On are the lengths of the ordinates

d = common distance

n = number of divisions

Note:

This rule is applicable only if ordinates are odd, i.e. even number of divisions.

If the number of ordinates are even, the area of last division maybe calculated separated and added to the result obtained by applying Simpson’s rule to two remaining ordinates.

Even if first or last ordinate happens to be zero, they are not to be omitted from Simpson’s rule.

The following offsets are taken from a chain line to an irregular boundary towards right side of the chain line.

Chainage 0 25 50 75 100 125 150
Offset ‘m’ 3.6 5.0 6.5 5.5 7.3 6.0 4.0

Common distance, d = 25m

Area = d/3[(O1+O7) + 2 (O3+O5)+4(O2+O4+O6)]

= 25/3[(3.6+4)+2(6.5+7.3)+4(5+5.5+6)]

Area = 843.33sqm

38 thoughts on “Methods of Calculation of Areas in Surveying | Simpson’s Rule”

  1. Hello,
    I am using Simpson’s rule in dredging quantity I am getting some deposition and dredging area i.e I am getting both negative and positive values
    Eg. Pre-level Post level Difference
    10.2 9.1 -1.1 (Pre level (-) Post level)
    11.2 10.2 -1.0
    10.6 11.0 0.4
    10.3 10.8 0.5
    like that, I am getting whether I can use both the signs in same Simpson’s rule.

    Please give me the solution as soon as possible. Thank you.

  2. Trapezpidal rule is based on the assumption that the several offset figures are trapezoids. That is basically the only assumption underlying the method that I know of .

    I must stumbling into this blog has been helpful thanks for your great work!

  3. Suppose, if ‘D’ at chainage 25 is ‘0’ in lieu of 5.0, can we still use Simpson’s Rule for calculating the area of the plot or we have to use any other method

  4. Suppose , if we want to calculate for different depth intervals ( Like 0- 1.5m depth and 1.5m – 3.0m depth) etc
    How to calculate volume using simpson’s rule.

  5. I am a Surveyor and Civil Engineer, this work is indeed very helpful for my research work, for practical purposes we should try to know what the total length is then fix it into a suitable interval that will be convenient. thanks.

  6. hi
    thnx s much 4 ur example it was really good.
    but plz could u send me more example which would be suitable for exam or testing students of technical institutes? could u help me plz. I real appreciate, I want some missing data in it or some thing atractives.
    thnx best regards

  7. hello..
    i am a marine engineer..i study naval architecture..
    all i know about simpson’s rule to find moments is that there has to be ‘odd’ ordinates to solve the problems using simpson’s multipliers…
    but recently i came across some unsolved questions which have even ordinates and i am really worried about how to solve them.
    please help me asap as such questions come in exams conducted by shipping ministry in india..
    thank you..

  8. Hi I used to calculate volume basically for dredging purpose using simpson first and second, for odd and even respectively A combination of first and third could give me a result to second order ploy, otherwise it will be a combination of second and third poly . I would like to derive third rule on third poly equation.

  9. Hi I am familiar with simpson first and second rule, can u give a detailed derivation of simpson third rule for calculating area between two leg if I know three I expect Integration

  10. i would like to know the limitation of simpson’s rule.
    and proper method for calculating the area of gujarat map? why ?

  11. i wanted to know if there is any condition in trapezoidal rule also like the one that ordinates should be odd in simpson’s rule

  12. I’m impressed, I need to say. Actually rarely do I encounter a blog that’s both educative and entertaining, and let me inform you, you could have hit the nail on the head. Your idea is excellent; the difficulty is something that not sufficient people are talking intelligently about

  13. I’m impressed, I need to say. Actually rarely do I encounter a blog that’s both educative and entertaining, and let me inform you, you could have hit the nail on the head. Your idea is excellent; the difficulty is something that not sufficient people are talking intelligently about. I am very happy that I stumbled across this in my search for one thing referring to this.

    • Hello,
      I am glad to have you visited my site. Thank you so much for your kind words of appreciation.
      Do keep visiting.

      Cheers:)

      • I have a doubt

        If we have Natural Ground Levels of a building (where cutting and Banking is coming),How can i Calculate the Average.N.G.L ?

  14. i think the formula for area of simpsons rule is quite wrong
    FROM THE FORMULA ABOVE –>
    area = d/3[(O1+O7) + 2 (O3+O5)+4(O2+O3+O6)]
    = 25/3[(3.6+4)+2(6.5+7.3)+4(5+5.5+6)]

    it should be –>
    area = d/3[(O1+O7) + 4 (O3+O5)+2(O2+O3+O6)]
    = 25/3[(3.6+4)+4(6.5+7.3)+2(5+5.5+6)]

    am i correct?? plez comment if i’m wrong..tq 🙂

    • Hello Oshin,
      Read the following statement carefully:

      Simpson’s Rule Statement

      It states that, sum of first and last ordinates has to be done. Add twice the sum of remaining odd ordinates and four times the sum of remaining even ordinates. Multiply to this total sum by 1/3rd of the common distance between the ordinates which gives the required area.

      As per the statement, the formula should be as follows:
      d/3(sum of first and last ordinate)+2(odd ordinates excluding the last ordinate and first ordinate) + 4(sum of remaining even ordinates)

      d/3(O1+O7) + 2(O3+O5) + 4(O2+O4+O6)

      Hope you got it…

      Cheers:-)

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