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Area Calculations
1
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.
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Division Into Simple Figures – cont.
 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.
Base x Height
Area =
2

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Triangle--cont.
 When the area forms an
equilateral or isosceles
triangle, determining
the height is not a
problem.
Area =
Base x Height
2
 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.
 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.
Area =
Base x Height
2
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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, Heron’s equation
can be used.
Area =
s s
- a s  b s  c 
a +b +c
s =
2
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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:
a x b x Sine 
Area =
2

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Division Into Simple Figures--Square & Parallelogram
The area of a square is determined
by:
Area = b x h
 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:
Area =  r2
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
 r2 
Area =
when the angle is known
360
compared to when the arc
length is known.

Area =
r x arc length
2

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Division Into Simple Figures --Trapezoid
 There are two different
trapezoidal shapes.
 The area equation is the
same for both.
Area = h x
a + b
2
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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.
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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 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 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.
X
Y
Plus
A
38.9
201.4
B
252.78
188.3
50,909.90
7,324.90
C
238.22
264.4
44,856.80
66,835.00
D
77.08
0
20,380.00
0.00
E
0
38.89
0.00
2,997.60
A
38.9
201.4
1,512.80
0.00
117,659.50
77,157.50
2
Minus
40,502.00
20,251.00 ft2
0.46 ac
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Coordinate Squares
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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 whole and
partial squares.
When the map scale is expressed as a
ratio, the area is determined by:

Area (ft ) = Grid size (in) x

2
2
Map scale (in)

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
Example: 1/2 inch grid is used and the
map scale is 1:1,000, then each square
would be equivalent to:

Area (ft ) = Grid size (in) x

2

= 0.5 x

2
Map scale (in)

12

2
1,000 

12 
= 1,736 ft2

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Coordinate Squares--cont.
If the map scale is expressed in in/ft then each grid area
is:
2
2
Area (ft )  Grid size x map scale

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:
Area (ft )  Grid size x map scale
2
2
 (0.50 in x 500)2
= 62,500 ft2

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Coordinate Squares
Example--cont.
Whole squares are counted.
Partial squares are estimated.
63 + 12 = 75 squares

ft2
Area
= Square size x
square

2
Map scale 

12

2

1000 
= 0.25 x

12 

ft2
434
x 75 square
square
Area (ac) =
ft2
43560
ac
32552.08 ft2
=
43560
0.75 ac
= 434 ft2/square

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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.
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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
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