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Part 1: Determining longitude and
latitude of a plot of land
(1)
Define the terms polygon and
regular polygon.
(3) Use a GPS
to determine the longitude and latitude of each vertex of the plot. Record
your coordinates.
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Example:
Below are the coordinates of the lot of Lambeth House at the
University of Virginia. These coordinates are used in the
spreadsheet screenshots on the cover page.
Vertex
1 W 78 30.591°
N 38 02.205°
Vertex 2
W 78 30.495°
N 38 02.155°
Vertex 3
W 78 30.525°
N 38 02.147°
Vertex 4
W 78 30.598°
N 38 02.188°
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(4) Compare your
approximations to your GPS readings.
Part 2: Converting differences in
longitude and latitude to linear distances
(5) Draw a sketch of your plot of land and label the
vertices in clockwise order. Discuss appropriate units of measurement for
the lengths of the sides and the area of this plot. Estimate the lengths
of the sides of the plot.
(6) Discuss how you would you convert differences in
latitude to linear distances and how you would convert differences in
longitude to linear distances. How would you calculate the linear
distances between vertices of your plot?
(7) Determine the number of miles per degree of
latitude and the number of miles per degree of longitude at the location
of your plot.
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Solution
1: Recall that lines of latitude are approximately 69 miles
apart. Hence, there are 69 miles per degree of latitude or 69/60
miles per second of latitude.
Solution
2: Note that lines of longitude are approximately 69 miles apart
at the equator but meet at the poles. Hence, there are 69 miles per
degree or 69/60 miles per second of longitude at the equator, but 0
miles at the poles.
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Solution
2 (continued): One way to determine the number of miles per
degree of longitude at your latitude is as follows: Draw a sketch of
the Earth using the Geometer’s Sketchpad. Place points to mark the
equator, the poles, and your location by latitude. The figure below
shows Charlottesville, VA at N 38.02°
latitude.
At any latitude, the number of miles per degree
of longitude equals the circumference of the Earth at that latitude
divided by 360. The ratio of the circumference around the Earth at
your location (e.g., G) to the circumference at the equator (B) is
equal to the ratio of the radii (i.e., segment IG and segment AB) at
those latitudes (why?). Set up the ratio between the radius of the
Earth at your latitude and that at the equator. Since there are 69
miles per degree of longitude at the equator, this ratio multiplied
by 69 will equal the number of miles per degree of longitude at your
location.
Note:
By moving point G on this
sketch you can determine the number of miles per degree of longitude
at any latitude. |
(8) Calculate the lengths of the sides of your plot
and record them on the sketch of your plot. Calculate the perimeter of
your plot.
Part 3: Determining areas of
polygons with Heron's Formula
(9) Discuss how to find the area of various polygons,
including those that are not regular. How would you find the area of an
n-sided polygon?
(10) Estimate the area of your plot. Discuss how you
would calculate the area of the plot.
(11) What is Heron’s formula?
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Heron’s
Formula: Heron’s Formula relates the area of a triangle to the
lengths of its sides. If a, b, and c are the lengths of the sides of
a triangle, then the area of the triangle is 
where s, the semi-perimeter, is equal to ½ (a+b+c).
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(12) Use Heron’s
formula to calculate the area of your plot.
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Note:
Subdividing your plot into triangles and applying Heron’s formula
to each triangular subdivision will yield the area.
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Part 4:
Generalizing the method to any convex polygonal plot of land
(13) Discuss how to generalize the method used to
calculate the area of your plot to any convex polygon.
(14) Create a spreadsheet where the input is a set of
coordinates for the vertices of any convex polygon and the output is the
area of the polygon.
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A
Solution: The spreadsheet on the cover page shows the
coordinates and area calculation for the lot of Lambeth
House at the University of Virginia.
The lot is a quadrilateral and is partitioned into two
triangles. With vertices input in clockwise order, the spreadsheet
first calculates the lengths of sides a,
b, and c
and the semi-perimeter s
for each triangle. It then uses Heron’s formula to calculate the
areas of each of the two triangles that partition this convex
polygonal area and then sums these areas to calculate the total
area. Each triangle uses a common “first point” as a vertex,
along with a “current point” and a “previous point.” See the
Sketchpad sketch and solution below for a description of the spreadsheet as
applied to a convex polygonal region subdivided into three
triangles.
Solution: The spreadsheet formula partitions
the area into triangles, all originating from the first vertex.
Going clockwise, a is calculated as the distance between the
first and previous vertices, b as the distance between the
previous and the current vertex, and c as the distance
between the current vertex and the first vertex. Entering the
formula in the spreadsheet this way allows us to fill down as more
vertices are added.
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Note 1:
In the spreadsheet, data were entered as minutes, in decimal form.
GPSs allow one to choose different data formats. The degree, minute,
second format requires that data be entered in the spreadsheet in
the [h]:mm:ss cell format, and then copied to cell with a
general format. This will display the readings as days and when
multiplied by 24 gives hours in decimal form. Hence it is easier to
record GPS data in a form ready for use in calculations.
Note 3:
There are 640 acres in a square mile.
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(15) Identify one or two
other accessible convex plots
of land. Estimate their areas. Use your GPS and spreadsheet to calculate
their areas.
Extension:
(16) Identify an
accessible concave plot of
land. Estimate the area of this plot. Discuss how you can use the GPS and
your spreadsheet to calculate the area of this plot. Calculate the area.
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