#### 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:

- Midpoint ordinate rule
- Average ordinate rule
- Simpson’s rule
- Trapezoidal rule
- 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 1/3^{rd}of the common distance between the ordinates which gives the required area.

**Where O _{1}, O_{2}, O_{3}, …. O_{n} 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[(O_{1}+O_{7}) + 2 (O_{3}+O_{5})+4(O_{2}+O4+O_{6})]

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

Area = 843.33sqm

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#1 by

Manoj Kumaron July 17, 2011 - 11:51 pmPlz 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…

#3 by

oshinon January 2, 2012 - 6:47 ami 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

BenzuJKon January 8, 2012 - 11:53 pmHello Oshin,

Read the following statement carefully:

Simpson’s Rule StatementIt 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:-)

#5 by

blue desperadoon April 8, 2012 - 9:46 amI’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.

#6 by

BenzuJKon April 10, 2012 - 1:11 amHello,

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 prasadon October 31, 2012 - 7:49 pmI 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 ?

#8 by

blue desperadoon April 8, 2012 - 9:46 amI’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

#9 by

rahulon April 16, 2012 - 9:26 pmi wanted to know if there is any condition in trapezoidal rule also like the one that ordinates should be odd in simpson’s rule

#10 by

HARDIK BHATTon April 17, 2012 - 9:22 pmi would like to know the limitation of simpson’s rule.

and proper method for calculating the area of gujarat map? why ?

#11 by

nurnabion May 21, 2012 - 10:28 pmur site is very help full for surveying student

#12 by

naizion June 8, 2012 - 1:02 amthanx

#13 by

bonifaceon June 22, 2012 - 1:25 amthink its the best site av seen so far thanx for the big help u r givin us so hw about the trapezeidal rule

#14 by

George Babyon September 19, 2012 - 8:46 pmHi 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

#15 by

George Babyon September 20, 2012 - 12:28 amHi 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.

#16 by

Aseemon November 4, 2012 - 6:41 amhello..

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..

#17 by

joneon November 5, 2012 - 10:56 amif the nb of ordinates is even what do i must do ?

#18 by

hassanon May 7, 2013 - 4:32 amif the interval of distance is not equal than what we are going to do?

#19 by

SHAILESH YADAVon October 16, 2013 - 10:30 pmIf we have only three ordinate can i simpson rule apply

#20 by

BenzuJKon October 22, 2013 - 7:45 amYes, you can use simpsons rule for three ordinates.

#21 by

BenzuJKon October 22, 2013 - 7:45 amYes. You can use simpsons rule for three ordinates.

#22 by

samiraon October 27, 2013 - 10:36 amhi

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

#23 by

mbisyo mon October 31, 2013 - 10:00 amwhat are the scenarios where you would use Simpson’s rule over the trapedzoidal rule

#24 by

ogwangericon November 20, 2013 - 11:34 pmam ogwang eric student of uganda technical college-elgon i have a problen in azimuth and bearing how can u help mi

#25 by

Edward Godwinon December 2, 2013 - 9:02 amWoow thanks, am glad i stumble into your side. God bless u

#26 by

clachrison December 20, 2014 - 2:28 amWanna know the calculations involved in simpson and trapezoidal rules

#27 by

peer basiton May 12, 2015 - 3:16 pmits just awesome i found my answer (y)

#28 by

Apim Victoron May 31, 2015 - 7:30 pmI 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.

#29 by

yeshiwondimon June 29, 2015 - 6:42 pmif the footing column movable what will happen to structure ? please send me answer

#30 by

BenzuJKon July 9, 2015 - 8:56 pmThe structure will collapse.

#31 by

Miki cengon September 10, 2015 - 1:55 pmCan you help in filling missed data in rise and fall method

#32 by

Yadanar Aungon March 11, 2016 - 12:29 pmHow shall i solve the problem if there is no odd ordinates by using simpson’s rule

#33 by

Mohanon April 11, 2016 - 5:25 pmSuppose , 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.

#34 by

Sakshion May 10, 2016 - 12:57 pmGood explanation

#35 by

Nirmala Adepon October 17, 2016 - 7:20 amSuppose, 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

#36 by

ALEKSANDRon November 6, 2016 - 2:18 pmTrapezpidal 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!