BASIC SURVEYING - AREAS AND VOLUMES COMPUTATION

AREAS AND VOLUMES

1. RGM COLLEGE OF ENGINEERING AND TECHNOLOGY
(AUTONOMOUS)
SUBJECT: SURVEYING
UNIT-III
COMPUTATION OF AREA AND VOLUME

2. Content:
•Introduction
•Methodsofcomputingarea
•Averageordinaterule
•Midordinaterule
•simpson’srule

3. INTRODUCTION
Incivilengineeringworksuchasdesignofbridges,dam
,reservoirsetc.Theareaofcatchmentofariverisrequired.Forroad
andrailwayslandistobeacquiredonthebasisofarea.Thus,finding
areasisessentialpartofsurveying.Itmaybenotedthattheareatobe
foundistheprojectedareauponthehorizontalplane.

4. unitsusedforfindingtheareaaresquare,meters,hectare,acres
etc.
100Sq.m=1are
100are=1hectare=10000Sq.m
1acre=4047Sq.m=2.5vigha
1vigha=16guntha
1acre=40guntha
1Hectare=2.471acres
1Sq.m=10000000Sq.m

5. COMPUTATION OF AREA FROM PLOTTED PLAN
•Boundary area can be
calculated as one of the
following rule:
–The mid-ordinate rule
–The average ordinate
rule
–The trapezoidal rule
–Simpson’s rule

6. METHODS OF COMPUTING AREA
Computationofareabytakingoffsets
1.Mid-ordinaterule
2.Averageordinaterule
3.Trapezoidalrule
4.Simpson’srule

7. MID-ORDINATE RULE
Inthismethodthebaselineisdividedintoa
numberofdivisionsandtheordinatesaremeasuredatthepoints
ofeachdivisions.Boundariesbetweentheoffsetsareconsidered
straightlines.

9. Whereh1,h2,h3,………… =midordinates
d=distanceofeachdivision
L=lengthofbaseline=nd
n=numberofdivision 
 
 
1 2 3
1 2 3
1 2 3
............
............
............
n
n
n
h h h h
Area L
n
h h h h
Area nd
n
Area h h h h d
   

   

     

10. AVERAGE ORDINATE RULE
Thisrulealsoassumesthattheboundariesbetweenthe
extremitiesoftheordinatesarestraightlines.

12. Whereh0,h1,h2,……=ordinatesofoffsets
d=distanceofeachdivision
n=numberofdivision
n+1=numberofoffsets
L=lengthofbaseline=nd 
 
1 2 3
1 2 3
............
1
............
1
n
n
h h h h
Area L
n
h h h h
Area nd
n
   


   



13. TRAPEZOIDAL RULE
Inthismethod,entireareaisdividedintotrapezoids.
Theruleismoreaccuratethantheprevioustworules.

15. whichisknownastrapezoidalrule.  
  
0 1 2 3 1
2
15
2 ..........
2
15
0 4.85 2 1.65 3.50 2.70 4.65 3.60 3.95
2
15 22.475
337.125
nn
dm
d
Area h h h h h h
Area
Area
Area m


       

        




16. Example:seriesofoffsetsweretakenfromachainlinetoan
boundary,intervalof15m,inthefollowingorder.
0,1.65,3.50,2.70,4.65,3.60,3.95,4.85m
Computetheareabytrapezoidalrule.
Solution:  
  
0 1 2 3 1
2
15
2 ..........
2
15
0 4.85 2 1.65 3.50 2.70 4.65 3.60 3.95
2
15 22.475
337.125
nn
dm
d
Area h h h h h h
Area
Area
Area m


       

        




17. SIMPSON’S RULE
Thisruleassumesthattheshortlengthsofboundary
betweentheordinatesareparabolicarcs.

19. Forsimpson’srule,thenumberofordinatemustbeodd.
simpson’sruleis:   
0 1 3 1 2 4 2
4 ........ 2 ..........
2
n n n
d
Area h h h h h h h h

         

20. APPLICATION:
•Simpson’s rule used for find the earthwork volume using
contour maps.it gives more accurate area.
•Trapezoidal rule can be applied for any number of ordinates. It
gives an approximate area
•A planimeter is used to measure the area of any shape with
more accuracy.
•Zero circle is used when the tracing point is moved , no
rotation of wheel will take place .

21. Example:Followingperpendicularoffsetsweretakenfroma
chainlineacurvedboundarylineatanintervalof10m.
0,7.26,5.83,6.45,7.20,8.18,8.0,0
computetheareabysimpsonsrule
Solution:
Tofindareabysimpson’srule,numberofoffsetsmustbe
odd.Herewehave8offsets.Therefore,foroffsetsh0toh6apply
simpson’sruleandforoffsetsh6andh7applytrapezoidalrule.

22. (continue)  
  
67
0 6 1 3 5 2 4
2
42
32
10 8 0
0 8 4 7.26 6.45 8.18 2 5.83 7.20 10
32
405.4 40
445.4
hhd
Area A h h h h h h h d
A
A
Am

        
 


       
 




23. COMPUTATION OF VOLUME
Content :
•Formulae for Calculation of Cross-Sectional Area
(a) Level Section
(b) Two level section
(c) Three Level Section
(d) Side Hill Two-Level Section
(e) Multi-Level Section

24. •Introduction:
•Forcomputationofvolumeofearthwork,thesectionalareaofthe
crosssectionwhicharetakentransversetothelongitudinalsection
duringprofilelevelingarefirstcalculated.Againcrosssectionmay
bedifferenttypesnamely…
(a) Level Section
(b) Two level section
(c) Three Level Section
(d) Side Hill Two-Level Section
(e) Multi-Level Section

25. •Themethodofcalculatingareasofsuchsectionsarecalculated
•Aftercalculationofcross-sectionalareas,thevolumeofearthwork
calculatedby…
•(a)thetrapezoidal(oraverageendarea)
•(b)theprismoidalrule
•Note:1.ThePrismoidalrulegivesthecorrectvolumedirectly
2.thetrapezoidaldoesnotgivethecorrectvolume.Prismoidal
correctionshouldbeappliedforthispurpose.This
correctionisalwayssubtractive.
3.cuttingisdenotedbyapositivesignandfillingbyaNegative
sign

26. Level Section :
When the ground is level along the transverse section

27. Example:Calculatethesectionalareaofanembankment10mwide
withasideslopeof2:1.Thegroundislevelinatransversedirection
tothecentreline.Thecentralheightoftheembankmentis2.5m
Here b = 10 m
s = 2
h = 2.5
Cross sectional area = (b + s X h)h
= (10+2 X 2.5)X2.5
= 37.5 m2

28. THANK YOU
Two-Level Section :
When the ground surface has transverse slope

30. Example:Thewidthoftheformationlevelofacertaincutting
is10mandthesideslopeis1:1.Thesurfaceofthegroundhas
auniformslopeof1in6intransversedirection.Letusfindthe
crosssectionalareawhenthedepthofthecuttingatthecenter
is3m