1 Area Calculations

2 Introduction Determining the boundaries and size of an area is a common occurrence. Chemical spill Wet land Watershed Etc.  For initial reports and estimates low precision methods can be used.  When a high level of accuracy is required, a professional engineer or a land surveyor should be employed.

3 Introduction--cont. Common methods of finding areas include: Division into simple figures Offsets form a straight line Coordinates Coordinate squares Digitizing coordinates Planimeter

4 Division Into Simple Figures

5  The area of of any polygon can be determined by dividing the field into simple figures and then calculating the area of each figure.  If completed using a map, the results will be less precise than when field measurements are taken.

Division Into Simple Figures – cont.  Common simple figures used are: Triangle Square/Rectangle Parallelogram Circle Sector Trapezoid 6

7 Division Into Simple Figures --Triangle  A triangle is three-sided figure or polygon whose interior angle sum is equal to 180 degrees.  Several different equations can be used to determine the area of a triangle.  Some triangle equations are easier to use than others, but the site will often determine which one that will be used because of the difficulty in collecting data.  The standard triangle equation is:  This is an easy equation to use, but measuring the boundaries can be difficult.  The difficulty is in measuring the height.

8 Triangle--cont.  When the area forms an equilateral or isosceles triangle, determining the height is not a problem.  Divide the base in 1/2 and turn a ninety degree angle at the mid point.

9 Triangle--cont.  Two types of triangles do not have two sides or two angles that are equal.  A triangle with three unequal lengths is called a scalene triangle.  A triangle with one angle greater than 90 degrees is called an obtuse triangle.  It is more difficult to determine the height for these triangles.

10 Triangle--cont.  The same equation can be used, the problem is determining the height.  When the area forms a scalene or obtuse triangle, the recommended procedure is to move along the base line and estimate where a perpendicular line intersects the apex of the triangle.  Turn a 90 degree angle and establish a line past the apex.  Measure the distance between the line and the apex (error).  Move the line the correct distance and direction along the base line and remeasure the height.

11 Triangle--cont.  It is not always desirable or practical to measure the height of a triangle.  An alternative is to measure the lengths of the sides.  When the lengths of the three sides can be measured, Heron’s equation can be used.

12 Triangle--cont.  There are occasions when neither the length of the three sides or the height of a triangle can be measured.  In this situation the area can be determined if one of the angles and the lengths of the two adjoining sides can be measured.  The equation is:

13 Division Into Simple Figures--Square & Parallelogram The area of a square is determined by:  The area for a parallelogram is determined using the same equation.  The difference is in how the height is measured. Measuring the height of a parallelogram is not as problematic as a triangle because the height can be measured at any point along the side by turning a 90 o angle and measuring the distance.

14 Division Into Simple Figures--Circle & Sector The standard area equation for a circle is: This equation is not practical for surveying because of the difficulty in finding the center of a circle.

15 Sector-cont.  A sector is a part of a circle.  A different equation is used when the angle is known compared to when the arc length is known.

16 Division Into Simple Figures --Trapezoid  There are two different trapezoidal shapes.  The area equation is the same for both.

17 Example Of Simple Figures  There is no right or wrong way to divide an irregular shape.  The best way is the method that collects the required information using the least amount of resources.

18 Area of Irregular Shape--cont. Which one of the illustrations is the best way to divide the irregular shaped lot?  The best answer? It depends. It is important to ensure all the figures are simple figures.

19 Offsets From A Line

20 Offsets From A Line Introduction  When a stream or river forms a property boundary, that side of the property will have an irregular edge.  Offsets from a line are used to form a series of trapezoids.  In this situation 90 o lines are established from the base line to a point on the irregular boundary.  The number of offsets and the offset interval is determined by the variability of the irregular boundary.

21 Offsets From A Line--cont. Each the area of each trapezoid is determined and summed to find the total area.

22 Area By Coordinates

23 Introduction  Determining area by coordinates is a popular approach because the calculations are easily done on a computer.  To determine the area, the coordinates for each corner of the lot must be determined. These can be easily determined using GPS. Coordinates can also be determined by traversing the boundary.

24 Coordinates--GPS  GPS equipment determines the location of points by one of two methods: Latitude & Longitude Universal Transverse Mercator (UTM)  Not very useful for determining areas.  Can be done, but complicated math. The UTM system determines the location of a point by measuring the distance east of a theoretical point and north of the equator. UTM measurements are easily used to determine area.

25 Coordinates By Traverse  A traverse is a surveying method that determines the boundary of an lot or field by angle and distances.  A traverse can be balanced to remove errors in measuring angles and distances.  The location of the corners can be converted to x - y coordinates.

26 Area By Coordinates Example  The first step is to determine the coordinates of each corner by establishing an x - y grid.  The math is easier if the grid passes through the southern most and western most point.  In this example UTM coordinates were used.  The next step is to set up a table to organize the computations.

27 Area By Coordinates Example--cont.  The area is computed by cross multiplying the X and Y coordinates and sorting them into the appropriate column.  The multiplication and sorting is controlled by a matrix.

28 Area By Coordinates Example--cont  After the matrix computations have been accomplished, the plus and minus columns are summed and subtracted.  The answer is divided by two.  This equals the area in square feet.

29 Area By Coordinates Example--cont.

30 Coordinate Squares

31 Coordinate Squares  This method overlays a map with a grid that has a known size.  Knowing the size of the grid and the scale of the map, the area can be determined by counting whole and partial squares. When the map scale is expressed as a ratio, the area is determined by: Example: 1/2 inch grid is used and the map scale is 1:1,000, then each square would be equivalent to:

32 Coordinate Squares--cont. If the map scale is expressed in in/ft then each grid area is: Example: a 1/2 inch grid is overlaid on a map with a scale of 1 in = 500 ft. The area of each grid is:

33 Coordinate Squares Example--cont. Whole squares are counted. Partial squares are estimated.

34 Coordinate Squares Example  Determine the area for the illustration.  The first step is to draw a grid on a clear material and lay it over the map. The area is determined by counting the grids.

35 Digitizing

36 Digitizing Coordinates  This method requires a machine called a digitizer.  The operator moves a special mouse or pen around the map and activates the mouse at each desired location.  Computer records x - y coordinates.

37 Planimeter

38 Planimeter  A Planimeter is a device the determines area by tracing the boundary on a map.  Two types: Mechanical Electronic

39 Questions