geomatic - areas & volume
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AREAS AND VOLUME
Mohd Efendi Daud (Dr. Sc)B.Surv (UTM, Malaysia) Msc (UTM, Malaysia), Dr. Sc (a!oya U"iv.,
#apa")
(Geomatic Division)
acu!t" o# $ivi! % Environmenta! En&ineerin&'Universiti un ussein Onn Ma!a"sia' *+,-- atu /ahat'
0ohor' MALA1S2A./hone 3 4+-5,6757+78 4+-9:5*675,-8 a; 3
4+-5,675-+-E<mai! 3 efendi=uthm.edu.m"
>e?3 htt@3BBB.#Cass.uthm.edu.m"
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2NRODU$2ON
• Area is often required in the context ofdesign, e.g., we might need the
surface area of a lake, the area of atract of land or of a cross-sectionalarea of road cutting.
• Volumes are often calculated by
integrating the area at regularintervals e.g., along a road centerline,or by using regularly spaced contours.
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• e need to know the volume ofmaterials in virtually all facets of civil
engineering especially in design andconstruction e.g., earthworkcomputations, amount of cut or !ll,amount of concrete, and etc.
• "here are two cases for calculating theareas#$ Areas of %traight &oundary
$ Area of 'rregular &oundary
2NRODU$2ON
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AREAS OSRA2G
OUNDAR1 • "he determination of areas depend
upon the topography of the terrain
and the accuracy required.• "he area of land portion, may be
determined by the following
methods#$ from the !eld notes, and
$ from the plotted plan or map
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$om@utation o# Areas #rom ie!dNotes
• henever the area of a plot of land is to bedetermined directly from the !eld notes, itshould be ensured that the survey lines includethe whole area and the land is divided into
geometrical !gures such as#-$ "riangles,
$ %quares,
$ (ectangles and etc.
AREAS OSRA2G
OUNDAR1
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AREAS OSRA2G
OUNDAR1 $om@utation o# Areas #rom ie!d Notes
( ) ( ) ( )c sb sa s s −−−
)ase *#
( )cba s ++=
2
1Area of triangle A&) is given aswhere,
= 0.5*ab*Sin C
= 0.5*ac*Sin B
= 0.5*bc*Sin A
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AREAS OSRA2G
OUNDAR1 $om@utation o# Areas #rom ie!d Notes
)ase +#
bh2
1=∆
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)ase #
AREAS OSRA2G
OUNDAR1 $om@utation o# Areas #rom ie!d Notes
ba×= "he area of a rectangle
S O
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AREAS OSRA2G
OUNDAR1 $om@utation o# Areas #rom ie!d Notes
)ase #
( ) d ba ×+=
2
1 "he area of a trapeium
AREAS O
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$om@utation o# Areas #rom /!ans
9. Gra@hica! method
AREAS OSRA2G
OUNDAR1
AREAS O
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$om@utation o# Areas #rom /!ans
. 2nstrumenta! method
AREAS OSRA2G
OUNDAR1
he /!animeter
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AREA 1$OORD2NAES
• "he area of shape A&)/0A can becalculated in the following
manner1111
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AREA O
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AREA O2RREGULAR
OUNDAR1 9. he ra@eoida! Ru!e
where,2*, 2+, 3 # 24set5 # 'nterval between o4set 6constant
A*A+ A A A8
AREA O
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9. he ra@eoida! Ru!e
AREA O2RREGULAR
OUNDAR1
"rapeoidal formula
A* 9 : 62* ; 2+7 5A+ 9 : 62+ ; 27 5A 9 : 62 ; 27 5 and,
A8 9 : 628 ; 2<7 5
F L (O9 4 O+ 4 (O 4O7 4 O,4 O6)H
AREA O
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. he Sim@sonIs Ru!e
2n Sim@sonIs Ru!e' it is assumed that the
irre&u!ar?oundar" is com@rised o# @ara?o!ic arcs.
AREA O2RREGULAR
OUNDAR1
Parabola arc
A F /7 (O9 4 O5 4 ,(O 4 O, 4 O+) 4 (O7 4O6)H
A F /7 9st
4 !ast ordinates) 4 ,
(even ordinates)4 (odd ordinates)H
AREA O
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. he Sim@sonIs Ru!e
AREA O2RREGULAR
OUNDAR1
Parabola arc
't is important to note that %impson=s (ule requires aeven num?er o# divisions o# the area, i.e. the tnum?er o# ordinates must ?e odd.
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VOLUMES
Mohd Efendi Daud (Dr. Sc)B.Surv (UTM, Malaysia) Msc (UTM, Malaysia), Dr. Sc (a!oya U"iv.,
#apa")
(Geomatic Division)
acu!t" o# $ivi! % Environmenta! En&ineerin&'Universiti un ussein Onn Ma!a"sia' *+,-- atu /ahat'
0ohor' MALA1S2A./hone 3 4+-5,6757+78 4+-9:5*675,-8 a; 3
4+-5,675-+-E<mai! 3 efendi=uthm.edu.m"
>e?3 htt@3BBB.#Cass.uthm.edu.m"
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2NRODU$2ON
• 'n civil engineering calculation of volumesof earthwork is required.
• "here are several methods available forthe determination of the volume of threedimensional !gures and earthworks.
• >enerally the methods are based on using
!eld survey techniques to acquire datathat represents the shape of the land sothat volume calculations can proceed.
• "he methods are as follow#
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VOLUMES
$ross<Sections
• A cross-section of the natural surface
indicates the existing ground pro!le normalto a !xed direction at a selection point.
• "he !xed direction is usually of signi!cancesuch as the center line of a road.
• 0xample of cross sections are described asfollow#
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$ross<Sections
• ?or a cut or !ll on horiontal ground
VOLUMES
$rea % h(&' "h)where h is the average cutting depth, " is the side slopeand ' is the formation width
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VOLUMES
$ross<Sections
• ?or a cut and !ll on sloping ground
Assuming a cut such as the ?ig.the cross-sectional area is found!rstly by calculating 5 @ >.
W L = S(b + nh)/(S + n)W G = S(b + nh)/(S-n)
Thus Area = (h '*")(W + W ) - '& *"
1: S & 1: n
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• * 9 g 6b+ ; hs7 6g-s7
• + 9 g 6b+ - hs7 6g-s7
• 5uas Bemotongan 9 6b+ ; gh7C 6g - s7D
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VOLUMES
Vo!umes #rom $ontours
contour *E+
contour *E 4
contour *E<
contour *EE
contour *FG
POINTS CONTOUR (m) AREAS (m )
A1 182 3150
A2 184 2460
A3 186 1630
A4 188 840
A5 190 210
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Vo!umes #rom $ontours < so!utions
VOLUMES
Trapezoidal formula
I = d [ (A1 + An) +A2 + A3 + A4 + .... + An -1
= d!2 [ (A1 + An) +2(A2 + A3 + A4 + .... + An-1)
= d!2 [ (A1 + A") +2(A2 + A3 + A4)
= 2!2 [ (31"# + 21#) + 2 ($4# + 1%3# + 24%#)
= 13, 220 m
Prizmoidal formula
I = d!3 [ (A1 + An) +4(A2 + A4 + &.. + An-1) + 2(A3 + A" )
= d!3 [ (A1 + A") +4(A2 + A4 ) + 2(A3)
= 2!3 [ (31"# + 21#) +4($4# + 24%# ) + 2(1%3#)
= 2!3 [ (33%#) +4(33## ) + 2(1%3#)
= 13,213.3 m
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Vo!umes #rom S@ot ei&hts
• "his method is particularly useful for large,open excavations such as tank, borrow pitsetc.
• "he area is divided into a grid, and levelsobtained at the intersection points.
• "he spacing of the grid depends on the terrain,accuracy required, and resources available.
VOLUMES
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Vo!umes #rom S@ot ei&hts
• "here are two methods$ (ectangular base, and
$ "riangular base
VOLUMES
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VOLUMES
(ectangular base - solution
Average 6/ug7 9 H.F*< 9 .<+Volume 9 Area x Average
9 6G x +87 x .<+ 9 <8.F m
+
Boints (educed 5evel/ug
6(educed 5ev/ug
5evel 6I72ccurance J
6J76J7 x 6I7
A *,.*8 *G.GG ,.*8 * ,.*8
& *,.HG *G.GG ,.HG + H.-G
) *-.,, *G.GG -.,, * -.,,
/ *,.F- *G.GG ,.F- + H.EE
0 *-.EG *G.GG -.EG - *F.+G
? *-.FH *G.GG -.FH + F.F-
> *8.*H *G.GG 8.*H * 8.*H
K *<.*G *G.GG <.*G + *+.+G
L *-.<H *G.GG -.<H * -.<H
*< H,.F-
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VOLUMES
"riangular base - solution
Boints (educed 5evel/ug
6(educed 5ev/ug
5evel 6I72ccurance J
6J76J7 x 6I7
A *.*8 *G.GG .*8 * .*8
& *.HG *G.GG .HG **.*G
) *. *G.GG . + E.<<
/ *.F *G.GG .F **.E+
0 *.EG *G.GG .EG < +E.EG
? *.FH *G.GG .FH *.F*
> *8.*H *G.GG 8.*H + *G.
K *<.*G *G.GG <.*G *E.G
L *.<H *G.GG .<H * .<H
+ ***.H8
Average 6/ug7 9 ***.H8+ 9 .<<Volume 9 Area x Average
9 6G x +87 x .<< 9 F+.+ m
+
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