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Area calculation

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**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

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Problem-1

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Problem 1-Result

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**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

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Mid-ordinate rule l

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Average ordinate rule

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Trapezoidal rule

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Simpson’s rule

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Problems The following perpendicular offsets were taken from chain line to an irregular boundary: Chainage m offset 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 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

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**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.

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d4/2 d4/2 d3/2 d3/2 D d m4 m3 4 3 Mid points Meridian distance of points (d1, d2, d3, d4) A c C 1 B 2 b d1/2 d1/2 d2/2 d2/2 m1 m2

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**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

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**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)

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

Area by Co-Ordinates

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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 method Side Latitude Departure N S E W AB 108 4 BC 15 249 CD 123 DA 257

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Volume calculation From cross sections From spot levels From contours

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

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**Methods of volume calculation**

Prismoidal method D2 A2 C2 B2 D2 D1 A2 A1 C1 C2 B1 B2 D1 A1 B1 C1

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**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}

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**A railway embankment is 10 m wide with side slopes 1. 5:1**

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 and the embankment has a rising gradient of 1:100. The ground is level across the centre line. Calculate the volume of earthwork. Distance 100 200 300 400 R.L. 204.8 206.2 207.5 207.2 208.3

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Area: Triangles and Trapezoids Lesson 10-2 p.509

Area: Triangles and Trapezoids Lesson 10-2 p.509

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