AreaAndVolume note for surveyor and the yard.

Area and volume note.

1. Area And Volume Calculation Er. Ganesh Prasad Sigdel Instructor LMTC

2. Area Would anyone tell me the area of Nepal? Correct! That’s 147181 sq. kilometers. Who can say?What does this nice digit 147181 mean?? Would this nice digit change, if I cut away all these mountains here in Nepal? WHY? OR, WHY NOT? 11/9/2014 2 Area And Volume,GP Sigdel, LMTC

3. AREA Some methods of determination of area of irregular figure: Using ordinates Average ordinate rule Mid-ordinate rule Simpson’s rule Trapezoidal Rule Using Absolute Coordinate Using Relative Coordinate By Total Lattitude And Sum of Departures By Double meridian distance and Lattitude By Graphical method Using Planimeter Using Computing Scale Using grids 11/9/2014 3 Area And Volume,GP Sigdel, LMTC

4. A no.of ordinates selected perpendicular to reference base line Average height of all ordinates taken Area calculated as : Average ordinate Rule 11/9/2014 4 Area And Volume,GP Sigdel, LMTC

5. h1 h2 h3 h4 h5 h6 h7 h8 hn d d d d Length Of base Average ordinate 11/9/2014 5 Area And Volume,GP Sigdel, LMTC

6. A no.of segments of equal width selected a vertical ordinate selected at center of each ordinate Assumption made that thus selected ordinate at the mid of division will have average height of two bounding heights of division. Area of each segment calculated as A=height of mid-ordinate *width of division Total area calculated from sum of all areas Mid -ordinate Rule h1 h2 h3 h4 h5 h6 hn d d d d d A1=h1 *d 11/9/2014 6 Area And Volume,GP Sigdel, LMTC

7. From figure below: Total Area=A1+A2+A3+…..+An =(h1*d)+(h2*d)+(h3*d)+……+( hn *d) =d (h1+h2+h3+………+ hn ) Hence, Area=Common distance(d) *sum of mid-ordinates Mid -ordinate Rule h1 h2 h3 h4 h5 h6 hn d d d d d A1=h1 *d 11/9/2014 7 Area And Volume,GP Sigdel, LMTC

8. Under-estimates ( concave upwards ) Over-estimates ( concave downwards ) Mid-ordinate Rule The following sketches show sample rectangles where the mid-ordinate rule under- and over-estimates the area. The blue shaded areas are not under the curve but are included in rectangle . The red shaded areas should be included but are not. 11/9/2014 8 Area And Volume,GP Sigdel, LMTC

9. Ordinates selected at uniform spacing Assumption: two ends of ordinates are connected by straight line so that each division takes form of a trapezium Area ore each trapezium calculated and summed up to get the total area of figure Trapezoidal Rule h1 h2 h3 h4 h5 h6 hn-1 d d d d d hn h7 A1= A2= A3= An= 11/9/2014 9 Area And Volume,GP Sigdel, LMTC

10. It seems from diagram below that the first and ordinate and last ordinate belong to only one trapezoid while others belong to two trapezoids. So, total area becomes: Trapezoidal Rule h1 h2 h3 h4 h5 h6 hn-1 d d d d d hn h7 A1= A2= A3= An= Trapezoidal Rule can be used in both cases when the no. of ordinates is odd or even Even if the first or last ordinate is zero, it needs to be included in formula as ordinate in calculation 11/9/2014 10 Area And Volume,GP Sigdel, LMTC

11. Considered as the most precise method of area calculation REASON:Assumptions consistent with real world situations Assumptions: End of ordinates are connected by arc of parabola The area of the part of parabola inside parallelogram is two- third the area of that of parallelogram Constraints And Necessary Conditions: No. of ordinates should be at least three No. of ordinates should be odd (or, no. of divisions should be oven) Simpson’s Rule Area=A Area=2A/3 11/9/2014 11 Area And Volume,GP Sigdel, LMTC

12. On derivation: If d be the common interval between ordinates, h1,h2, h3, h4, h5, ……..hn-1, hn be the height of ordinates: Simpson’s Rule 11/9/2014 12 Area And Volume,GP Sigdel, LMTC

13. Trapezoidal Vs. Simpson’s Rule Concave Towards Base Convex Towards Base Area from Simpson’s Rule Greater than Area from Trapezoidal Rule Area from Simpson’s Rule less than Area from Trapezoidal Rule 11/9/2014 13 Area And Volume,GP Sigdel, LMTC

14. Find the area of the following figures using: Average ordinate rule Simpson’s rule Trapezoidal rule Some example questions 15 08 07 12 11 7 8 16 24 32 40 10 20 30 40 55 70 100 85 9 17 10 10 8.5 14 13.5 14.5 8.5 08 07 12 11 7 6 12 18 24 30 36 11/9/2014 14 Area And Volume,GP Sigdel, LMTC

15. Area From Independent Coordinates Procedure of method List co-ordinates each vertices in an order, either clockwise or anti-clockwise Repeat the first co-ordinate once again Perform cross multiplication- with upward multiplication positive and downward diagonal These coordinates should be placed inside modulus sign to avoid negative value of area Half of the sum of this product is the area of the figure E1,N1 E2,N2 E3,N3 E4,N4 11/9/2014 15 Area And Volume,GP Sigdel, LMTC

16. Area from Independent Coordinates E1,N1 E2,N2 E3,N3 E4,N4 Positive product Negative product A= 11/9/2014 16 Area And Volume,GP Sigdel, LMTC

17. 150,150 175,200 150,225 100,200 200,175 Positive product Negative product A= Find the area of a pentagon whose vertices, taken in order are (150,150),(200,175),(175,200),(150,225),(100,200) 11/9/2014 17 Area And Volume,GP Sigdel, LMTC

18. Two methods commonly used: Area from lattitude and double meridian distance Area from Total lattitude And Sum of departures Area from Relative Coordinates 11/9/2014 18 Area And Volume,GP Sigdel, LMTC

19. Area from double meridian distance and lattitude Meridian distance of a line : the perpendicular distance (i.e. along East direction) from meridian to mid point of that line Double meridian distance of a line: sum of meridian distances of its two ends Lattitude of a line: length of projection of line along meridian Departure of a line: length of projection of line perpendicular meridian E N A B C D E MD of line AB Departure of AB Lattitude of AB 11/9/2014 19 Area And Volume,GP Sigdel, LMTC

20. Area from double meridian distance and lattitude Procedure: Arbitrary meridian assumed to pass throught the westmost point of the closed figure The westmost point is that where the departure changes its sign from negative to positive Then DMD of each line determined with following rules: The DMD of the first line is equal to it’s departure DMD of any other line= DMD of previous line+departure of previous line+departure of that line DMD of last line= it’s departure with positive sign E N A B C D E MD of line AB Departure of AB Lattitude of AB DMD of each line is then multiplied by it’s lattitude Then the half of the sum of (DMD times lattitude )of all sides is equal to the total area of figure 11/9/2014 20 Area And Volume,GP Sigdel, LMTC

21. 150,150 175,200 150,225 100,200 200,175 Find the area of a pentagon whose vertices, taken in order are (150,150),(200,175),(175,200),(150,225),(100,200) A B C D E Arithmetic Checks The sum of lattitudes should be zero The sum of departures should be zero The DMD of last line should be equal to positive value of its Departure 11/9/2014 21 Area And Volume,GP Sigdel, LMTC

22. Area from Total lattitude and sum of departures Statement: Twice the area within the lines of the closed traverse is equal to the algebric sum of products of total lattitude of each station by the algebraic sum of lines meeting at that station . Total lattitude of a point is the total distance from the reference E-W line, taken parallel to the meridian E N A B C D E Lattitude of BC Total Lattitude of point C 11/9/2014 22 Area And Volume,GP Sigdel, LMTC

23. Area from Total lattitude and sum of departures Statement: Twice the area within the lines of the closed traverse is equal to the algebric sum of products of total lattitude of each station by the algebraic sum of lines meeting at that station . Total lattitude of a point is the total distance from the reference E-W line, taken parallel to the meridian E N A B C D E Lattitude of BC Total Lattitude of point C 11/9/2014 23 Area And Volume,GP Sigdel, LMTC

24. Area from Total Lattitude and sum of departures Procedure: Arbitrary E-W line assumed to pass throught the southmost point of the closed figure The southtmost point is that where the lattitude of line changes its sign from negative to positive With reference to that meridian, net Northing or total meridian distance of each point is determined as: Total meridian distance of any point=Total meridian distance of previous point+ lattitude of line joining these two points ( eg . Total Lattitude of E= Total lattitude of D + lattitude of DE) At each station, the sum of departures calculated by adding departures of two lines meeting at that point. Thus calculated sum of departure at each station multiplied with total lattitude of corresponding station Half the sum of these products yield the area of that closed figure 11/9/2014 24 Area And Volume,GP Sigdel, LMTC

25. 150,150 175,200 150,225 100,200 200,175 Find the area of a pentagon whose vertices, taken in order are (150,150),(200,175),(175,200),(150,225),(100,200) A B C D E 11/9/2014 25 Area And Volume,GP Sigdel, LMTC

26. Determine the area of the following closed traverse using a) Independent coordinates b)Total lattitude and sum of departures c)DMD and lattitude 11/9/2014 Area And Volume,GP Sigdel, LMTC 26 Side Lattitude Departure AB 40 30 BC -20 45 CD -30 10 DE -20 -50 EA 30 -35 Side Departure Lattitude PQ -35 15 QR 25 RS 55 10 SP -20 -50 Side Departure Lattitude JK -20 -25 KL 10 -30 LM 25 -15 MN 20 55 NJ -35 15

27. 11/9/2014 Area And Volume,GP Sigdel, LMTC 27 AREA BY GRAPHICAL METHOD

28. Area By planimeter Planimeter , instrument to measure area on a plan drawn on paper Formed of: Anchor arm with anchor point and tracing arm with tracing point Area measured on the disc( or dial), roller (or wheel) and vernier 11/9/2014 Area And Volume,GP Sigdel, LMTC 28

29. Structure of a planimeter Anchor: point to support planimeter to hold rigidly at a point Anchor arm: a bar with one end fixed at the anchor end and other end attached to integrating unit Tracing arm: arm with tracing point at one end, connected with anchor arm at hinge Tracing point: Point with pin to trace/ follow the periphery of plan to be measured Integrating unit: consists of disc, wheel and the vernier. 11/9/2014 Area And Volume,GP Sigdel, LMTC 29

30. Structure of a planimeter On moving tracing arm, the wheel rotates and reading is obtained Drum attached to wheel graduated into 100 divisions Attached to drum is vernier that reads 1/100 th of division of wheel Dial divided in two parts. 1 complete revolution of wheel increases reading on wheel by 1. Reading taken as: unit on dial, 1/10 th and 1/100 th on wheel and 1/1000 th on vernier So if values are: 3 on dial, 43 on wheel drum and 8 on vernier, the reading is3.438 11/9/2014 Area And Volume,GP Sigdel, LMTC 30

31. Computing Area With Planimeter Anchor point selected inside or outside the area to be measured, so that its boundaries on all sides can be traced with tracing point. A point marked on boundary and tracing point placed over it Initial reading recorded by observing wheel, disc and vernier The tracing point moved along the boundary till it returns back to starting point The final reading observed and area computed by using formula. 11/9/2014 Area And Volume,GP Sigdel, LMTC 31

32. 11/9/2014 Area And Volume,GP Sigdel, LMTC 32 Computing Area With Planimeter

33. Generally provided by the manufacturer If not provided can be calculated as: 11/9/2014 Area And Volume,GP Sigdel, LMTC 33 Constant of Planimeter / Multiplying Constant

34. Any circle has area But some circle, whose area on being traced by planimeter may be found as zero Called zero circle Cause: only sliding motion of wheel occurs, no rotating motion at all 11/9/2014 Area And Volume,GP Sigdel, LMTC 34 Zero Circle

35. VOLUME 11/9/2014 Area And Volume,GP Sigdel, LMTC 35

36. From cross-sections From Spot Levels From Contours 11/9/2014 Area And Volume,GP Sigdel, LMTC 36 Measurement of Volume

37. Level section: cross section with both of it’s its top surface on constant elevation or on a level section Side slope: The inclination of the sloping line with respect to vertical line generally expressed as S:1 (horizontal distance: Vertical distance) Taken along the cross section, i.e. perpendicular to the direction of alignment 11/9/2014 Area And Volume,GP Sigdel, LMTC 37 Volume Measurement from cross section (level section)

38. Top of the section needs to be a level surface This type of section observed in case of canal Slope on both sides are equal and uniform section is assumed throughout the portion If side slope be s:1, shorter base be b and height be hm from figure, length of longer side becomes b+2sh. 11/9/2014 Area And Volume,GP Sigdel, LMTC 38 Volume Measurement from cross section (level section) b h b 1 s sh sh b+2sh

39. This type of section appears in case of embankment Slope on both sides are equal If side slope be s:1, base be b and height be h.from figure, length of shorter side or top of embankment becomes b+2sh. 11/9/2014 Area And Volume,GP Sigdel, LMTC 39 Volume Measurement from cross section (level section) h b 1 s sh sh b-2sh

40. Requires parallel sections of some regular of irregular cross-sections Areas do not need to be equal or similar, but cross sections need to be parallel and uniformly spaced Assumes that area between cross sections change linearly. 11/9/2014 Area And Volume,GP Sigdel, LMTC 40 Volume By Trapezoidal/End Area Formula d d d d d A1 A2 A3 A4 A5 An-1 An

41. Considered as the most precise method of volume computation No. of cross-sections necessarily needs to be odd Areas do not need to be equal or similar, but cross sections need to be parallel and uniformly spaced Assumes that cross sections are connected by arc of parabola. 11/9/2014 Area And Volume,GP Sigdel, LMTC 41 Volume By Prismoidal /Simpson’s Formula d d d d A1 A2 A3 A4 A5 An-1 An d

42. Volume computed by trapezoidal formula is usually greater than that by Prismoidal formula An exception is that when in a section, center height/area is higher with narrower base and second section has lower center height/area with wider base Volume obtained from Prismoidal formula is the precise one A correction applied to the area obtained from Trapezoidal Formula to convert it to equivalent of Prismoidal formula, called Prismoidal Correction Prismoidal correction is negative/ subtractive The error in trapezoidal formula depends upon the difference between successive areas. Larger the difference, higher the error The prismoidal correction can be upto 16.66 percent of the trapezoidal area 11/9/2014 Area And Volume,GP Sigdel, LMTC 42 Prismoidal Correction

43. Above formulas based on the assumption that cross- sections are parallel But in curves, cross- sections are are taken in radial lines In such case, correction needs to be applied to the volume. And the volume is calculated using ?? Pappu’s Theorem 11/9/2014 Area And Volume,GP Sigdel, LMTC 43 Curvature correction in Volume Computation

44. “Volume swept by a constant area rotating about a fixed axis is equal to the product of the area and the length of path traced by it’s centroid .” Curvature correction is: Positive when the centre of curve and centroid of area lie on opposite side of centerline Negative when center of curve and centroid of area lie on same side of centerline Curvature correction for a level section is zero (Refer to Surveying And Levelling by R. Agor , Chapter 9 Volumes for derivation, if you are eager on it) 11/9/2014 Area And Volume,GP Sigdel, LMTC 44 Curvature Correction And Pappu’s Theorem

45. Principle: Connecting three points on the surface of earth forms a truncated prism. The volume of truncated prism is equal to its base area times the average height from design elevation Similarly , the area of Truncated Rectangular prism with points A, B, C and D will be : 11/9/2014 Area And Volume,GP Sigdel, LMTC 45 VOLUME FROM SPOT HEIGHT A C B ha hb hc

46. Procedure Divide the total area into a no. of equal sized triangles’ network On doing so, each vertex may contribute to 1 to 8 triangles Calculate total height of cutting and multiply it by the area of single truncated prism, that gives the total earthwork 11/9/2014 Area And Volume,GP Sigdel, LMTC 46 VOLUME FROM SPOT HEIGHT

47. VOLUME FROM SPOT HEIGHT 11/9/2014 Area And Volume,GP Sigdel, LMTC 47

48. 11/9/2014 Area And Volume,GP Sigdel, LMTC 48 Volume From Contour