Finding the Areas of shapes in graphs

 In a graph, there are numerous points which join to form various polygons. These polygons may be irregular in shape, with any number of sides. The area of these polygons can be found using different methods!


If we have a regular polygon, we can use that polygon's formula to find the area. So if we have a hexagon, we can just use the formula :


                                    a^2(3√3)/2

This is the formula to find the area of a regular hexagon. 'a' represents the length of a side. There is a different formula for different regular polygons. 

The side length of a polygon can be found using the distance formula, where we calculate the distance between two points.


But, what if the polygons are not regular? If they have different side lengths and different angles on each side? The formula generally used here is commonly known as the shoelace method!

The shoelace method works for regular and irregular shapes, and one only needs the coordinates of the points of the polygon. 



We arrange the coordinates in this way, so if the polygon we want to measure the area of is a pentagon, 'n' will be 5, as there will be 5 sets of coordinates. We solve this in steps:

The arrows imply multiplication. We multiply the values and store the sum of the blue arrows and the sum of the red arrows separately. 

=> x1y2+x2y3+x3y4.....xny1 (this is the sum of the products of all the values of the red arrows)

=> y1x2+y2x3+y3x4.....ynx1 (this is the sum of the products of all the values of the blue arrows)

Now we subtract the blue arrows from the red arrows. Let's take a variable, p, to simplify

=> p= red arrows - blue arrows

                       or 

=> p=( x1y2+x2y3+x3y4.....xny1)-(y1x2+y2x3+y3x4.....ynx1 )

If p is negative, we just change the sign!

Now we divide by two to get the final area!


Worked example:

Lets take the coordinates as:

(1,7)

(2,3)                                       

(3,5)




=> 3+10+21=34 (sum of all red arrows)

=> 14+9+5=28 (sum of all blue arrows)

=>34-28=6

=>6/2=3

The area of this polygon is three!

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