How to find the area of an irregular shape on a grid

Introduction

I just thought I would share with you a clever technique I once used to find the area of general polygons.

The polygon could be regular (all angles are equal and all sides are equal) or irregular

The Example Polygon

Let's use this polygon as an example:

Coordinates

The first step is to turn each vertex (corner) into a coordinate, like on a graph:

Area Under One Line Segment

Now, for each line segment, work out the area down to the x-axis.

So, how do we calculate each area?

Average the two heights, then multiply by the width

Example: For the shape highlighted above, we take the two heights (the "y" coordinates 2.28 and 4.71) and work out the average height:

(2.28+4.71)/2 = 3.495

Work out the width (the difference between the "x" coordinates 2.66 and 0.72)

2.66-0.72 = 1.94

The area is width×height:

1.94 × 3.495 = 6.7803

Add Them All Up

Now add them all up!

But the trick is to add when they go forwards (positive width), and subtract when they go backwards (negative width).

If you always go clockwise around the polygon, and always subtract the first "x" coordinate from the second, it works out naturally, like this:

You can also go the other direction. If you get a negative area just make it positive.

And it looks like this:

So that's it! The area is 8.3593

Area of Polygon Tool

Glad you read this far! You are rewarded with a link to the Area of a Polygon Drawing Tool that can do all of this for you. It also accepts manual entry of coordinates.

The area of irregular shapes means the space occupied by the shape which is measured in square units. An irregular shape can be of any size and length. Irregular shapes can be seen all around us, for example, a kite, a diamond shape, a leaf, etc. Any shape whose sides and angles are not of equal length is termed an irregular shape.

What is the Area of Irregular Shapes?

The area of irregular shapes is the amount of region covered by that shape. Irregular shapes are those shapes that do not have equal sides or equal angles. In order to find the area of an irregular shape, it can be decomposed or divided into multiple familiar shapes and then the area of those can be added to get the total area. We can see different examples of irregular shapes in our everyday life:

• Observe the staircase of a building. The surface area of the staircase is an irregular shape composed of polygons like rectangles and squares.
• A playground in a school with a running track is an irregular shape and is also a combination of regular shapes.
• The leaf of a plant is an irregular shape.

The unit of area of an irregular shape is expressed in terms of m2, cm2, in2, or ft2.

How to Find the Area of Irregular Shapes?

In order to find the area of irregular shapes, we must know how to decompose an irregular shape into familiar shapes.

Decompose the Irregular Shape

The following shape is an irregular figure whose area cannot be calculated by a specific formula. If we decompose it, we can see that it can be split into polygons P, Q, R, S, T, and U. In other words, these polygons join together to form the irregular shape. So, after decomposing, we find the area of these polygons and add the values to get the area of the given irregular shape.

Let us take another example. The given figure can be split into multiple polygons A, B, C, D, E, F, and G. Since we recognize these shapes, we can easily find the area of these and add them to get the area of the given irregular shape.

Formula for Finding the Area of Irregular Shapes

There is no direct formula to calculate the area of an irregular shape. We use different methods to find the area in such cases.

Method 1: As we discussed earlier, to find the area of irregular shapes, we can divide the irregular shape into familiar polygons and then find the area of each individual polygon.

Therefore, the area of the given irregular shape = Area of all the polygons that form the irregular shape.
⇒ Area of the given irregular shape = area of A + area of B + area of C + area of D + area of E + area of F + area of G

Method 2: In some cases, the irregular shape is divided into small unit squares or grids. The total number of unit squares falling within the shape determines the total area. It should be noted that while calculating the area for a more accurate estimate we count the square as “1” if the shaded region covers more than half of each square. So, in this case we get an approximate area of the irregular shape. We can use this method for shapes which have curves like the one shown in the following figure.

Therefore, the area of the given irregular shape = Sum of all the unit squares falling under the irregular shape.

FAQs on Area of Irregular Shapes

What is the Area of Irregular Shapes?

The area of irregular shapes is defined as the amount of space that is covered by an irregular shape. Irregular shapes are those shapes that do not have equal sides or equal angles. The unit for the area of an irregular shape is expressed in terms of square units, for example, m2, cm2, in2, or feet2.

How is Area of Irregular Shapes Different From Regular Shapes?

A regular shape has equal sides and angles, whereas, an irregular shape can be of any size and length. Thus, the area of regular shapes can be determined by directly applying suitable formulas for it whereas the area of an irregular shape can be found by decomposing an irregular shape into several regular shapes.

Is there a Formula to Find the Area of Irregular Shapes?

There is no specific formula to find the area of an irregular shape since an irregular shape has all sides of different lengths. Sometimes, an irregular shape is made up of a combination of polygons, thus, the formula is determined by decomposing an irregular shape into regular shapes.

What is the Method to Find the Area of Irregular Shapes?

We can find the area of irregular shapes using the following steps:

• Step 1: Decompose an irregular shape into familiar polygons.
• Step 2: Find the sum of all the areas of the known polygons to find the area of the irregular shape.

How Can We Find the Area of Irregular Shapes Using the Unit Square Method?

We can find the area of irregular shapes using the unit square method with the following steps:

• Step 1: Divide the given irregular shape into small squares of the unit square area.
• Step 2: The total number of unit squares falling within the shape determines the total area.
• Step 3: While calculating the area for a more accurate estimate count the square as “1” if the shaded region covers more than half of each square.

Is the Unit Square Method an Accurate Way to Find the Area of Irregular Shapes?

No, while measuring the area of an irregular shape if we use the unit square method, we get an approximate value of the area instead of the accurate value.