AREA METHODS

1. AREA METHODS Often a need to calculate areas from existing plans or from measurement taken on site.

2. Areas from existing plans • Counting square • Bigger regular shapes and give and take lines • Planimeter – mechanical integration

3. Areas from site measurement • Trapezoidal formula • Simpsons formula • Coordinate method

4. 1. Counting squares • Divide into equal squares • Count whole squares • Estimate part squares (use additional grid to estimate the fraction) • Sum whole squares and part squares • Calculate area of one square ensuring square measured using appropriate scale. • Total area = area of one square x sum of squares 0.8 0.95 0.9 1 1 1 1 1 1 1 1

5. 2. Give and take lines • Simplify shape with give and take lines • Measure resultant simple shapes • Sum areas of all shapes

6. Simplify shape Simplify shape with straight lines

7. Ensure “Give = take”

8. Measure sides of simple shapes

9. c a b Use standard formulas Area of triangle = s(s-a)(s-b)(s-c) Where s= (a+b+c)/2 • Simplify shape with give and take lines • Measure resultant simple shapes • Sum areas of all shapes

10. 3. Planimeter • Mechanical device or more recently electronic devices which converts perimeter length to an area. • Very difficult to use and must measure at least twice to check results

11. Planimeter – use • The brass cylinder is anchored to the table with a point, like a compass point. It pivots, but does not slide. The elbow joint bents and slides freely. The pointer on the other end is used to trace the perimeter of the region. Near the elbow is a wheel, which simply rolls and slides along the tabletop. The scale is on the wheel itself, so it tells how far the wheel has turned. Sure enough, that number is proportional to the area of the region. The conversion factor depends on the scale of the drawing or photograph. • The scale wheel is attached to the "green" arm, near point B, and its axis is parallel to the green arm, BC. This orientation is important. Suppose that the green arm has a translational motion. That is, it slides but it is always pointed in the same direction. If it moves longitudinally, then the wheel will not turn at all. It will merely slide sideways. If the arm moves in any other direction, then the rotation of the wheel will be proportional to the component of the translation that is normal to BC. Also, it would not matter where the wheel is attached, as long as it stays fixed to the green arm, with its axis parallel to BC.

12. D Y1+YN + n= N-1 Yn { D } Area = n  Y 2 n=2 4. Trapezoidal formula • Typically used with offset measurement taken as part of site survey. • Area = Strip Width x {Average of first and last offset + Sum of the rest of the offsets}

13. Area = Y0 +YN 2 Yodd + 4 Yeven + D 3 { } 5. Simpson rule • Typically used with offset measurement taken as part of site survey • More accurate than Trapezoidal method • Odd number of Offsets only • Area = 1/3 Strip width x {First + Last Offset +4 x Sum of even Offsets +2 Sum of Odd Offsets}

14. 6. Coordinate Method • Typically used to find area within Traverse Survey. • Coordinates of all stations making up the survey used. • Ideal method to use with computer spreadsheets.

15. 1 2 En{Nn-1 - Nn+1} Area = Coordinate method -2 E7, N7 E1, N1 • Best computed using a tabular approach • Spreadsheet ideal approach • Compute clockwise otherwise negative results E6, N6 E2, N2 E5, N5 E4, N4 E3, N3

16. Area by Coordinates -2