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Using a GPS Handheld Receiver to Determine Distance
and Area of Polygonal Plot

 Activity Description Activity Guide

Part 1: Determining longitude and latitude of a plot of land

(1)   Define the terms polygon and regular polygon.

(2) Identify a nearby plot of land that has a polygonal shape and accessible vertices (a field at your school, your house lot, etc.). Roughly approximate the longitude and latitude of this plot of land. Discuss how you made these approximations.

(3) Use a GPS to determine the longitude and latitude of each vertex of the plot. Record your coordinates.

 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°

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.

 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.

 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.

 Hint: Use the number of miles per degree of latitude and the number of miles per degree of longitude (at your latitude) as “conversion factors” in the distance formula. The formula in the spreadsheet on the activity description page uses a conversion factor of 69/60 miles for each second of latitude and 54.35/60 for each second of longitude at N 38.02° latitude to calculate the lengths of the sides of the lot at Lambeth House.

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?

 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).

(12) Use Heron’s formula to calculate the area of your plot.

 Note: Subdividing your plot into triangles and applying Heron’s formula to each triangular subdivision will yield the area.

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.

 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.

 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 2: This calculation does not account for the curvature of the Earth. For small distances the curvature can be ignored. Note 3: There are 640 acres in a square mile.

(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.

 Hint: If your convex polygon has one “concave” vertex, the spreadsheet described above will work if the concave vertex is chosen as the “first point” common to all triangular subdivisions.

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