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



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  1. #1 by Manoj Kumar on July 17, 2011 - 11:51 pm

    Plz check the formula again as it doesnt match with your example
    Regards
    RKM

    • #2 by BenzuJK on July 18, 2011 - 7:04 am

      Hello Manoj,
      I checked with the formula for Simpson’s rule. Its absolutely right. Check your calculations…

  2. #3 by oshin on January 2, 2012 - 6:47 am

    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 :)

    • #4 by BenzuJK on January 8, 2012 - 11:53 pm

      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:-)

  3. #5 by blue desperado on April 8, 2012 - 9:46 am

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    • #6 by BenzuJK on April 10, 2012 - 1:11 am

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

      Cheers:)

      • #7 by Devi vara prasad on October 31, 2012 - 7:49 pm

        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 ?

  4. #8 by blue desperado on April 8, 2012 - 9:46 am

    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

  5. #9 by rahul on April 16, 2012 - 9:26 pm

    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

  6. #10 by HARDIK BHATT on April 17, 2012 - 9:22 pm

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

  7. #11 by nurnabi on May 21, 2012 - 10:28 pm

    ur site is very help full for surveying student

  8. #12 by naizi on June 8, 2012 - 1:02 am

    thanx

  9. #13 by boniface on June 22, 2012 - 1:25 am

    think its the best site av seen so far thanx for the big help u r givin us so hw about the trapezeidal rule

  10. #14 by George Baby on September 19, 2012 - 8:46 pm

    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

  11. #15 by George Baby on September 20, 2012 - 12:28 am

    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.

  12. #16 by Aseem on November 4, 2012 - 6:41 am

    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..

  13. #17 by jone on November 5, 2012 - 10:56 am

    if the nb of ordinates is even what do i must do ?

  14. #18 by hassan on May 7, 2013 - 4:32 am

    if the interval of distance is not equal than what we are going to do?

  15. #19 by SHAILESH YADAV on October 16, 2013 - 10:30 pm

    If we have only three ordinate can i simpson rule apply

    • #20 by BenzuJK on October 22, 2013 - 7:45 am

      Yes, you can use simpsons rule for three ordinates.

    • #21 by BenzuJK on October 22, 2013 - 7:45 am

      Yes. You can use simpsons rule for three ordinates.

  16. #22 by samira on October 27, 2013 - 10:36 am

    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

  17. #23 by mbisyo m on October 31, 2013 - 10:00 am

    what are the scenarios where you would use Simpson’s rule over the trapedzoidal rule

  18. #24 by ogwangeric on November 20, 2013 - 11:34 pm

    am ogwang eric student of uganda technical college-elgon i have a problen in azimuth and bearing how can u help mi

  19. #25 by Edward Godwin on December 2, 2013 - 9:02 am

    Woow thanks, am glad i stumble into your side. God bless u

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