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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.

€

Area = Base x Height

2

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Triangle--cont.

When the area forms an equilateral

€

Area = Base x Height

2

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.

The same equation is 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.

€

Area = Base x 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.

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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, Heron’s equation can be used.

€

Area = s s - a( ) s − b( ) s −c( )( )

s =a + b + c

2

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:

€

Area = a x b x Sine θ

2

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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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Area = b x h

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Circle

The standard area equation for a circle is:

€

Area = π r 2

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

Circle

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

Circle - radius

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One method uses chords and perpendicular lines.

How effective would this be?

Circle – cont.

A more practical equation for the area of a circle uses the diameter.

How do you determine the diameter of a circle?

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€

Area=π D2

4

One method uses the greatest distance.

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

€

Perimeter=π D

D =Perimeter

πHow do you measure the perimeter of a circle?

Sector

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

€

Area =π r 2 θ360

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Area = r x arc length

2

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.

€

Area = h xa + b

2

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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.

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.

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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 - +

A 38.90 201.40

B 252.78 188.30 7324.87 50909.89

C 238.22 264.40 66835.03 44856.83

D 77.08 0.00 0 20379.95

E 0.00 38.89 2997.64 0

A 38.90 201.40 0 1512.82

Sum 77,157.540 117,659.490

Difference 40,501.950 (Double area)

Divided by 2 20,250.975 Square Feet

0.46 Acres

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Map Methods

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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 squares. Whole and partial

squares are counted.

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

€

(Area ft 2) = ( ) Grid size in x ( )Map scale in

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⎛

⎝ ⎜

⎞

⎠ ⎟2

Example 1/2 inch grid is used and the map scale is 1:1,000, then each square would be equivalent to:

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(Area ft 2) = ( ) Grid size in x ( )Map scale in

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⎛

⎝ ⎜

⎞

⎠ ⎟2

= 0.5 x1, 00012

⎛

⎝ ⎜

⎞

⎠ ⎟2

= 1, 736 ft 2

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Coordinate Squares--cont.

If the map scale is expressed in in/ft then each grid area is:

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(Area ft 2) = Grid size x map scale( )2

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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(Area ft 2) = Grid size x map scale( )2

=(0.50 500)in x 2

= 62, 500 ft 2

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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.

€

63 + 12 = 75 squares

Areaft 2

square= Square size x

Map scale12

⎛

⎝ ⎜

⎞

⎠ ⎟2

= 0.25 x100012

⎛

⎝ ⎜

⎞

⎠ ⎟2

= 434 ft 2/ square

€

( )Area ac =434

ft 2

square 75 x square

43560 ft 2

ac

=32552.08 ft 2

435600.75 ac

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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?