area calculation. area is divided into triangles, rectangles, squares or trapeziums area of the one...

Area calculation

• Area is divided into triangles, rectangles, squares or trapeziums• Area of the one figure (e.g. triangles, rectangles, squares or

trapeziums) is calculated and multiplied by total number of figures.• Area along the boundaries is calculated as

• Total area of the filed=area of geometrical figure +

boundary areas

Problem-1

Problem 1-Result

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

Mid-ordinate rule

l

Average ordinate rule

Trapezoidal rule

Simpson’s rule

Problems• The following perpendicular offsets were taken from chain line to an

irregular boundary:– Chainage 0 10 25 42 60 75 m– offset 15.5 26.2 31.8 25.6 29.0 31.5

Calculate the area between the chain line, the boundary and the end offsets.

• The following perpendicular offsets were taken from a chain line to a hedge:

Calculate the area by mid ordinate and Simpson’s rule.

chainage 0 15 30 45 60 70 80 100 120 140

offsets 7.6 8.5 10.7 12.8 10.6 9.5 8.3 7.9 6.4 4.4

• Area by double meridian distances– Meridian distance of any point in a traverse is

the distance of that point to the reference meridian, measured at right angle to the meridian.

– The meridian distance of a survey line is defined as the meridian distance of its mid point.

– The meridian distance sometimes called as the longitude.

12

34

A

B

C

D

Mid pointsMeridian distance of points (d1, d2, d3, d4)

d1/2 d1/2 d2/2 d2/2

d3/2d3/2d4/2d4/2

m1

m2

m3

m4d

c

b

• Meridian distances of survey line:– m1=d1/2– m2= m1+d1/2+d2/2– m3=m2+d2/2-d3/2– m4=m3-d3/2-d4/2

• Area by latitude and meridian distance– Area of ABCD=area of trapezium CcdD + area of

trapezium CcbB – area of triangle AbB – area of triangle AdD

– = m3*L3+ m2*L2-1/2*2*m4*L4-1/2*2*m1*L1

– =m3*L3+m2*L2-m4*L4-m1*L1

• Double meridian distance:– M1= meridian distance of point A + meridian

distance of point B

– M1=0+d1

– M2=meridian distance of point B + meridian distance of point C

– =d1+(d1+d2)

– =M1+(d1+d2)

– M3=(d1+d2)+(d1+d2-d3)

• Area of the traverse ABCD = M3*L3+M2*L2-M1*L1-M4*L4

• Area by Co-Ordinates

• The following table gives the corrected latitudes and departures (in m) of the sides of a closed traverse ABCD. Compute the area by (a) M.D. method (b) co-ordinate methodSide Latitude Departure

N S E W

AB 108 4

BC 15 249

CD 123 4

DA 0 257

Volume calculation

• From cross sections

• From spot levels

• From contours

Measurement of volume

• 3 methods generally adopted for measuring the volume are– (i) from cross sections– (ii) from spot levels– (iii) from contours

Methods of volume calculation

A1

A2

B2

D2

C2

B1

D1

C1

A1

A2

B2B1

D2

C2

D1

C1

• Prismoidal method

• Also called Simpson’s rule for volume.

• Necessary to have odd number of cross sections.

• What if there are even number of C/S?

• Trapezoidal method:– Assumption mid area is mean of end areas.

– Volume =d{(A1+An)/2+A2+A3+…+An-1}

• A railway embankment is 10 m wide with side slopes 1.5:1. assuming the ground to be level in a direction transverse to the centre line, calculate the volume contained in a length of 120 m, the centre heights at 20 m intervals being in metres 2.2, 3.7, 3.8, 4.0, 3.8, 2.8, 2.5.

A railway embankment 400 m long is 12 m wide at the formation level and has the side slope 2:1. The ground levels across the centre line are as under:

The formation level at zero chainage is 207.00 and the embankment has a rising gradient of 1:100. The ground is level across the centre line. Calculate the volume of earthwork.

Distance 0 100 200 300 400

R.L. 204.8 206.2 207.5 207.2 208.3