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Areas and VolumesIntroduction
Faculty of Applied Engineering and
Urban Planning
Civil Engineering Department
2nd Semester 2008200!
S u r v e y i n g
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Areas andVolumes
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Planimeters
Digital Planimeter "ptical Polar Planimeter
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Planimeters
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Questions?!
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Areas
1. Regular Figures
• Mathematical Formulae
•Method o oordinates
". Irregular Figures
• #ra$hical Method
• %ra$e&oidal Rule
• Sim$son's (ne)%hird Rule
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Mathematical Formulae
sin(C) ba2
1or bh
2
1⋅⋅ 2a ba ⋅
( ) bah2
1
+⋅ °
⋅⋅ n180
cotan4
12
2
dπ4
1
⋅
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Mathematical Formulae
( )21
2
2 r r π −
( ) baπ ⋅ bh3
2
2r π
360
1⋅∆⋅
∆−∆ sin180
πr
2
1 2
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Mathematical Formulae
15 m20 m
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Questions?!
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Areas *y Method o oordinates
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Areas *y Method o oordinates
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Areas *y Method o oordinates
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Areas *y Method o oordinates
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Areas *y Method o oordinates
Area # 0$%&' ()*8+0$0( , &-8+8!0$!+. '.
# ((08*%$+8 ft2
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Questions?!
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Areas o Irregular Figures
1. #ra$hical Method
". %ra$e&oidal Rule+. Sims$on's (ne)%hird Rule
ontent
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Trapezoidal ule
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Trapezoidal ule
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Trapezoidal ule
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!xample
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"imspon#s $ne%Third ule
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!xample
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%ra$e&oidal Rule
&rea 'alculation
Sims$on's (ne)%hird Rule
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• Volumes *y3
1. Average)nd)Area Method
". Prismoidal Method
+. ontour Ma$s
• Volume rom S$ot -evels
ontent
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'ut and (ill
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Average)nd)Area Method
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Prismoidal Method
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alculating Volumes rom ontour Ma$
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alculating Volumes rom ontour Ma$
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alculating Volumes rom ontour Ma$
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Questions?!
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Volume rom S$ot -evels
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Volume rom S$ot -evels
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Mass )aul *iagram
A mass aul diagram is of great value bot in planning
and construction$
1e diagram is plotted after te eartor3 4uantitiesave been
computed5 te ordinates soing
aggregate volumes in cubic metres ile te
ori6ontal base line5 plotted to te same scale as te
profile5 gives te points at ic tese volumes
obtain$
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Mass )aul *iagram
7ost materials are found to increase in volume after
ecavation &9bul3ing9.5 but after being re-compacted byroller
or oter means5 soils in particular migt be found
to occupy less volume tan originally5 i$e$ a 9srin3age9
as ta3en place en compacted in te in situ volume$
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Defnitions
+- Haul refers to the volume of material multiplied by
thedistance moved, expressed in ‘station meters'.
+- Station meter (stn m) is 1 m3 of material moved 1
m, Thus, / m0 mo1ed 2// m is a haul of / 3 2//4// 50// stn
m.
+0- !aste is the material excavated from cuts but not used
foremban"ment #lls.
+6- $orro% is the material needed for the formation
ofemban"ments, secured not from road%ay excavation but
from elsewhere. 7t is said to be obtained from a 8borrow
pit#.+2- &imit of economical haul is the maximum haul
distance.
!hen this limit is reached it is more economical to wasteand
borrow material.
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Bulking and shrinkage
!xca1ation of material causes it to loosen,and thus its
exca1ated 1olume will begreater than its in situ 1olume.
)owe1er,
when lled and compacted, it may occupya less 1olume than when
originally in situ.
(or example, light sandy soil is less byabout 9 after lling,
whilst large rocs
may bul by up to 6/9. To allow for this, a correction
factor is
generally applied to the cut or ll 1olumes.
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Construction o the MHD
& M)* is a continuous cur1e, whose1ertical ordinates,
plotted on the samedistance scale as the longitudinal
section, represent the algebraic sum ofthe corrected 1olumes
+cut ;, ll
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Mass )aul *iagram
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Mass )aul *iagram
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