### Shoelace Formula

There are many different formulas that can use to calculate area of simple polygon whose vertices are described by their Cartesian coordinates in the plane. The *s**imple polygon is defined as a flat shape consisting of straight, non-intersecting line segments or “sides” that are joined pair-wise to form a closed path. *Basic formula uses **Shoelace formula** (also known as Gauss’s area formula and the surveyor’s formula). The Shoelace formula gets its name because if one lists the coordinates in a column, and marks the pairs of coordinates to be multiplied, the resulting image looks like laced-up shoes.

### Grid to Ground Area

The scale factors consist of elevation scale factor and grid scale factor were explained in detailed at the “*Point Scale Factor*” and “*Line Scale Factor*“.

The method to convert from grid-based area to ground-based area needs two steps same as the distance conversion, the first step divided the square of **grid scale factor** (GSF) from grid-base area which the result is ellipsoidal area, the second step deriving the ground-based area by divided the ellipsoidal area with the square of **elevation scale factor** (ESF). Often these scale factors are combined in what is known as a **Combined Scale Factor** (CSF) which allows ground-based area to be calculated from grid-based area in one step. The grid-based area can be converted to ground-based area by divided grid-based area with the square of the combined scale factor. The effect of scale factors sometimes cannot be negligible In extreme cases such as in urban areas where property value is high, the impact of neglecting to account for scale factors can have a significant financial impact on owners of property rights.

#### Grid to Ellipsoidal Area by Projection of Lambert Azimuthal Equal Area

Another method to convert the grid-based area to ellipsoidal area by using the projection of Lambert Azimuthal Equal Area (LAEA) instead of divided by the square of grid scale factor. We know that all the Equal Area projections preserve the area measure, generally distorting shape. The Lambert azimuthal equal-area projection is a particular mapping from a sphere to a disk (that is, a region bounded by a circle). It accurately represents area in all regions of the sphere, but it does not accurately represent angles. Take a look at the images that illustrate a cross sectional view of the sphere and a plane tangent to it at *S*. Each point on the sphere (except the antipode) is projected to the plane along a circular arc centered at the point of tangency between the sphere and plane.

If the user provides the projected coordinates of the polygon to be calculated the area. The “**Area**” tool will tangent the planar to the **centroid** of the area, then transform the coordinates to the LAEA projection, therefore the results of the calculation will be received the new set of projected coordinates but on LAEA projection. Finally the tool will calculate the ellipsoidal area from LAEA coordinates by Shoelace’s formula. The last step need to convert the ellipsoidal area to ground-based area by divided the ellipsoidal area with the square of elevation scale factor (ESF).

### The Area tool

When you start the Surveyor Pocket Tools at the main window you will see the “Area” tool as the following.

Start to run the “Area” tool and the window will be displayed as below.

The screenshot below illustrate the components of the “Area” tool.

#### Example Data

In this tutorial use example data that bundled with the installation. If you are doubtful where is the file and folder. Please follow the link below.

How to locate and use example data?

#### Example 1 – Test on Lao National Datum 1997

The example file to be demonstrated is located on the “Example folder” and will use the file name “*boundary7-lao-lao1997-utm48n.csv*“.

Given:

The coordinate system : Lao National Datum 1997

Ellipsoid : Kassovsky 1940

Datum transformation Parameters : DX** =** +46.012 meters, DY = -127.108 meters and DZ** = **-38.13**1** meters

Select the coordinate system to “*Lao National Datum 1997/ UTM zone 48N*“

Click the button. The small dialog will be populated. Click “*Browse…*” to browse the CSV file.

Browse to “Example folder” and select “*boundary7-lao-lao1997-utm48n.csv” *and open.

The small dialog will be displayed the contents of the CSV file, then click “OK” to continue.

The contents of CSV file will be read into table view as below. If you notice the column settings all the columns display “*Col1*” that mean all of them are not configured yet.

Change the column settings to the corresponding value as below.

Select “*Vertical Reference*” to “*Orthometric Height (H)*” and enter the value to “*1352*“. Please note the unit is meter.

Click the button to compute the areas.The results from the calculation are shown ground-based area, ellipsoidal area and original grid-based with the different values due to the scale factor effect.

#### Area Comparison

The results of area computation as above example are listed as the followings:

- Grid-based Area =
**12041.416**m² (calculated from the coordinates of CSV file) - Ellipsoidal Area =
**12024.398**m² (calculated from the coordinates of LAEA projection on WGS84 datum) - Ground-based Area =
**12029.376**m² (calculated by divided the ellipsoidal area with the square of elevation scale factor)

The coordinates of the centroid of this area was calculated and the results value are *21.680704936N, 102.100947415E* in the terms of latitude and longitude on WGS84 datum. This coordinate values will be displayed later when export to Microsoft Excel file. Calculate the grid scale factor and transform the coordinates to origin grid coordinate system based on “Lao National Datum 1997” by **Transform Coordinates** tool.

The grid scale factor (GSF) at the centroid area = *1.0007121270. *Calculate the ellipsoidal area by formula:

Ellipsoidal Area = Grid-based Area / GSF²

Ellipsoidal Area = 12041.416 / 1.0007121270² = 12024.284 m²

Compare the value 12024.284 m² to **12024.398 **m² that calculated from the coordinates of LAEA projection on WGS84 datum. Two values are a little bit different. Using the LAEA projection will get the more accurate area than use the formula as aformentioned above due to the grid scale factor of the centroid point is only the average grid scale factor of this area.

Use the **Point Scale Factor** tool to calculate the average elevation scale factor (ESF) from the centroid point.

Ground-based Area = Ellipsoidal Area / ESF² = 12024.398 / 0.9997930752² = 12029.376 m²

#### Export to Microsoft Excel

Click the “Export…” button to export the Microsoft Excel file.

The Save file dialog will be populated. Browse to the folder that you want to save the file. The next step select the file *type* to save. There are three choices in the “Save as type:”: *Excel, CSV and ESRI Shape file.*

Select “*Excel (*.xlsx)*” and enter the file name.

#### Open the file with Spreadsheet Application

Open the saved file with the spreadsheet application, in my case I use **LibreOffice Calc**.

The Excel file consists of four sheets:

*Ground-based area*sheet contain the columns coordinate of LAEA projection and two columns of cross-multiplying illustrate the Shoelace’s formula.*Grid-based are*a sheet contain the columns of original grid coordinates and the column of cross-multiplying as well.*Projection*sheet contain the information of Lambert Azimuthal Equal Area projection.*Ellipsoid*sheet contain the ellipsoid parameter.

The next screenshot below shows the “Grid-based area” sheet.

Scroll down to the bottom of the sheet. Check the cell of the sheet, the sheet contains the Shoelace’s formula using the spreadsheet formula to compute cross-multiplying and net summation and finally the result is grid-based area.

Look at the “*Projection*” sheet, there are information of the **Lambert Azimuthal Equal Area** projection:

*Latitude of Origin*uses latitude value of centroid point.*Central Meridian*uses longitude value of centroid point.*False Northing*and*False Easting*use zero value.*Ellisoid*uses “WGS84”.

Take a look at “*Ground-based area*” sheet, scroll down to the bottom of the sheet.

Please note the ellipsoidal area was calculated by the spreadsheet formula which generated by the “**Area**” tool. Finally the ground-based area was computed by divided the ellipsoidal area with the square of elevation scale factor (ESF).

#### Export to Shape file

Click the “*Export…*” button and browse to your folder and select the “*Save as type:*” to “*ESRI Shape file*” then enter the file name.

Test the file with the QGIS, overlay with Google Satellite by WMS service.

#### Save KML file and Display on Google Earth

Click the button to save KML file and display the polygon to Google Earth. The cyan marker shows the *Ground-base area* as the screenshot below.

#### Example 2 – Test on WGS84 Geodetic Coordinate

Even if the tool designed for projected coordinate system but it can apply for the geodetic coordinate as well. This example demonstration will use the file name “*boundary4-crossed-zone47n-zone48n.csv*” that located on “*Example folder*“. Click the button to open the file.

Given:

- Coordinate System : “Geographic / WGS84”
- Average elevation (Orthometric height(H)) : 34.512 m

Enter the elevation on “*Orthometric Height (H)*” text box and click button to calculate the area.

Even if the input CSV file coordinates are based on WGS84 geodetic coordinate system but the tool computed the *Grid-based Area* on WGS84 / UTM as well. Please note the difference of the grid-based area and ground-based area is about *840 square meter* that too much to ignore. Pin the polygon to Google Maps as the screenshot below.

### Conclusion

The “**Area**” tool can calculate the grid-based area from the CSF file and convert to true ground-based area with the ease of use.

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