Area Calculations

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Area Calculations
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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.

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Introduction--cont.

• Common methods of finding areas include
• Division into simple figures
• Offsets form a straight line
• Coordinates
• Coordinate squares
• Digitizing coordinates
• Planimeter

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Division Into Simple Figures
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Division Into Simple Figures

• 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.
• Common simple figures used are
• Triangle
• Square/Rectangle
• Parallelogram
• Circle
• Sector
• Trapezoid

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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.

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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.

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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.

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Triangle--cont.

• 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.

• The same equation can be used, the problem is
  determining the height.

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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, Herons equation can be used.

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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

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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 90o angle and measuring the distance.
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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.
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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.
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Division Into Simple Figures --Trapezoid

• There are two different trapezoidal shapes.
• The area equation is the same for both.

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Example Of Simple Figures

• There is no right or wrong way to divide the
  irregular shape.
• The best way is the method that requires the
  least amount of resources.

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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.

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Offsets From A Line
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Offsets From A LineIntroduction

• 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 90o 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.

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Offsets From A Line--cont.

• Each the area of each trapezoid is determined and
  summed to find the total area.

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Area By Coordinates
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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.

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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.

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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.

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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.

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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.

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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.

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Area By Coordinates Example--cont.
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Coordinate Squares
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Coordinate Squares

• When the map scale is expressed as a ratio, the
  area is determined by

• 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.

Example 1/2 inch grid is used and the map scale
is 11,000, then each square would be equivalent
to
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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
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Coordinate SquaresExample--cont.

• Whole squares are counted and then partial
  squares are estimated.

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Coordinate SquaresExample

• 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.
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Digitizing
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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.

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Planimeter
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Planimeter

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

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Questions