Area Calculations

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Area Calculations
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Introduction

• Determining the size of an area is a common
  problem of landscaping.
• Application of chemicals
• Amount of sod
• Number of acres
• Etc.
• Minimum skill required

• When a high level of accuracy is required, a
  professional engineer or a land surveyor should
  be employed.
• Area for legal document
• Etc.

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Introduction--cont.

• Areas are calculated by using one of two
  methods.
• Field measurements
• Map measurements
• Common field measurements for determine area
  include
• Division into simple figures
• Offsets form a straight line
• Coordinates

• Common map measurements for determining area are
• Coordinate squares
• Division into simple figures
• Digitizing coordinates
• Planimeter

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Field Measurements
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Division Into Simple Figures

• The area of complex shapes can be determined by
  dividing the field into simple figures and then
  calculating the area of each figure.
• Common simple figures used are
• Triangle
• Square/Rectangle
• Parallelogram
• Circle
• Sector
• Trapezoid

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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.
• The best equation to use is determined by the
  site.
• 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 the same.
• A triangle with no equal lengths or angles is
  called a scalene triangle.
• A triangle with one angle greater than 90 degrees
  is called an obtuse triangle.

• It can be 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 is used, the problem is
  determining the height.

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Triangle - cont.

• Because of obstructions or other limiting factors
  alternative methods may be necessary.
• When it is not possible to traverse the interior
  of an obtuse or scalene triangle this method can
  be used.

• Note for this method to work the height must be
  measured perpendicular to the baseline.

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Triangle--cont.

• It is not always possible to measure the height
  of a triangle.
• When the lengths of the three sides can be
  measured, Herons equation can be used.

Note many times the perimeter is the easiest
measurement. Setting this up on a spreadsheet
eliminates all the calculations.
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Triangle--cont.

• There are occasions when neither the length of
  one side nor 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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Square Parallelogram

• A square is a simple figure where all four sides
  and all four angles are equal.
• 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.

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Circle

• The standard area equation for a circle is

• This equation works well in math class, but how
  do you find the radius of a circle?

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Circle

• To understand the following methods you must know
  the parts of a circle.

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Circle - radius

• One method uses chords and perpendicular lines.

• How effective would this be?

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Circle cont.

• A more practical equation for the area of a
  circle uses the diameter.
• How do you determine the diameter of a circle?

• One method uses the greatest distance.

• Another method uses the perimeter of a circle and
  the perimeter equation.

• How do you measure the perimeter of a circle?

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Sector

• A sector is a part of a circle.
• Two equations can be used.

• When the angle is know the area is a proportion
  of a circle.

• When the radius and arc length can be measured
  the arc length is used.

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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,
  one side of the property will have an irregular
  edge.
• 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.
• This method results in a series of trapezoids.

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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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Area By GPS Coordinates

• GPS equipment determines the location of points
  by one of two methods
• Latitude Longitude
• Universal Transverse Mercator (UTM)
• Latitudes and longitudes are angles referenced
  from Greenwich Mean and the equator.
• 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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Area 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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Traverse cont.

• A traverse survey must be balanced to account for
  measuring errors.
• Balancing a traverse requires several sequential
  steps that must be done correctly.

DEC LAT DEP COR COR BAL LAT BAL DEP COR
STA N/S DEG MIN SEC E/W FT ANG COS SIN - - LAT DEP - - DIST

• Computer programs are available to do this.

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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.
Sta x y y -
A 38.90 201.40 201.40
B 252.78 188.30 188.30 7324.87 50909.89
C 238.22 264.40 264.40 66835.03 44856.83
D 77.08 0.00 0.00 0 20379.95
E 0.00 38.89 38.89 2997.64 0
A 38.90 201.40 201.40 0 1512.82
Sum Sum 77,157.540 117,659.490
Difference Difference Difference 40,501.950 (Double area)
Divided by 2 Divided by 2 Divided by 2 20,250.975 Square Feet
0.46 Acres
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Map Methods
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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
  squares.
• Whole and partial squares are counted.

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

• Determine the area for the illustration.
• The first step is to draw a grid on clear
  material and lay it over the map.

• The area is determined by counting the grids.

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Coordinate SquaresExample--cont.

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

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Simple Figures

• The simple figures method works the same for both
  field and map methods.
• In the map method a scale is used to measure the
  distances from the map.
• It is easier to determine the distances from a
  map than to measure them out in the field.

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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

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

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Questions?