Area of an Irregular Polygon
Unlike a regular polygon, unless you know the coordinates of the vertices, there is no easy formula for the area of an irregular polygon.
Each side could be a different length, and each interior angle could be different. It could also be either convex or concave.
If you know the coordinates of the vertices of the polygon, there are two methods:
- A manual method. See Area of a polygon (Coordinate geometry).
- A computer
See Algorithm to find the area of any polygon
So how to do it?
One approach is to break the shape up into pieces that you can solve - usually triangles,
since there are many ways to calculate the area of triangles.
Exactly how you do it depends on what you are given to start.
Since this is highly variable there is no easy rule for how to do it.
The examples below give you some basic approaches to try.
1. Break into triangles, then add
In the figure on the right, the polygon can be broken up into triangles by drawing all the diagonals
from one of the vertices. If you know enough sides and angles to find the area of each, then you can simply add them up to find the total.
Do not be afraid to draw extra lines anywhere if they will help find shapes you can solve.
Here, the irregular hexagon is divided in to 4 triangles by the addition of the red lines.
( See Area of a Triangle)
2. Find 'missing' triangles, then subtract
In the figure on the left, the overall shape is a regular hexagon, but there is a triangular piece missing.
We know how to find the area of a regular polygon so we just subtract the area of the 'missing' triangle created by drawing the red line.
(See Area of a Regular Polygon and
Area of a Triangle.)
3. Consider other shapes
In the figure on the right, the shape is an irregular hexagon, but it has a symmetry that lets us break it into two parallelograms
by drawing the red dotted line. (assuming of course that the lines that look parallel really are!)
We know how to find the area of a parallelogram so we just find the area of each one and add them together.
(See Area of a Parallelogram).
As you can see, there an infinite number of ways to break down the shape into pieces that are easier to manage.
You then add or subtract the areas of the pieces.
Exactly how you do it comes down to personal preference and what you are given to start.
4. If you know the coordinates of the vertices
If you know the x,y coordinates of the vertices (corners) of the shape, there is a method for finding the area directly.
See Area of a polygon (Coordinate geometry). This works for all polygon types (regular, irregular, convex, concave). There is also a computer
that does the same. See Algorithm to find the area of any polygon
Related polygon topics
Types of polygon
Area of various polygon types
Perimeter of various polygon types
Angles associated with polygons
(C) 2009 Copyright Math Open Reference. All rights reserved
Math Open Reference now has a Common Core alignment.
See which resources are available on this site for each element of the Common Core standards.
Check it out