Area on a Coordinate Grid
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Finding the area of polygons drawn on a coordinate axis is an easy process.   There are two situations to be considered when examining these polygons:


Sides are parallel to the axes.

If the figure is drawn such that its sides (or needed segments) are drawn ON the grids of your graph paper, you can COUNT the lengths and use your area formulas.


Sides are NOT parallel to the axes.

If the figure is drawn such that its sides (or needed segments) are NOT drawn ON the grids of your graph paper, you will need to draw a "BOX" around the figure to determine its area.

 

Sides are parallel to the axes:

COUNT
to find the needed lengths.

In this example, the base of the triangle lies on the grid of the graph paper, and the altitude also lies on the grid of the graph paper.

To COUNT:  stand at A and take one step to the right to the next grid line.  Continue stepping and counting until you reach C.

From counting, we know the base is 6 and the altitude is 3.

The area of a triangle formula:

You could also find the length from A to C by subtracting the x-coordinates of the two points.    4-(-2) = 6

For the altitude, you need to determine that the base of the altitude is (2,1).  Then subtract the y-coordinates of the two points.  
 4 - 1 = 3.

The answer is 9 square units.
 

 

Sides are NOT parallel to the axes.


 ** Find the area of the "box" by counting.

** Represent the triangle you wish to find by x.

"Box" Method
to find area.
In this example, the sides of the triangle do NOT lie on the grid of the graph paper.  You should:

1.  Draw the smallest "box" possible which will enclose the polygon (in this case a triangle).  Be sure the "box" follows the grids of the graph paper.

2.  Number each of the parts of the box with a Roman numeral (ignore the coordinate axes when numbering). 

3. "The whole is equal to the sum of its parts."  The area of each of the parts added together equals the area of the "box".
    

 
** Find the the area of each of the right triangles by
 counting and using the formula for the area of a triangle.
 
      
    The answer is 12 square units.

 
 

Dealing with odd shaped pieces.

Further subdivide ...
There are times when the parts of the "box" will not form nice right triangles in each of the corners.  Notice the upper left hand corner of the "box" in this example.  It was necessary to further subdivide that upper left section into one square and two right triangles, so that the lengths could be counted easily..

Remember to keep your work simple by forming shapes that are easy to count.

Follow the same procedure that was developed in the previous problem.


 

The answer is 33 square units.

 


It's easy, but it's tiring!