# 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 6.5 5.5 7.3 6 4

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. How can I find out the earth cutting and filling volume, I have the contour survey

3. 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!

4. 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

5. Good explanation

6. 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.

7. How shall i solve the problem if there is no odd ordinates by using simpson’s rule

8. Can you help in filling missed data in rise and fall method

9. if the footing column movable what will happen to structure ? please send me answer

• The structure will collapse.

10. 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.

11. its just awesome i found my answer (y)

12. Wanna know the calculations involved in simpson and trapezoidal rules

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

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

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

16. 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. If we have only three ordinate can i simpson rule apply

• Yes, you can use simpsons rule for three ordinates.

• Yes. You can use simpsons rule for three ordinates.

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

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

20. 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.
thank you..

21. 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.

22. 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

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

24. thanx

25. ur site is very help full for surveying student

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

27. 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

28. 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

29. 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 ?

30. 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,

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:-)

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

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

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