Area of Irregular Shapes

Calculating the area of irregular shapes can seem challenging at first glance, as they do not have defined formulas like squares or circles. However, several methods can help in accurately finding the area of these complex shapes. This article will guide you through different techniques for finding the area of irregular shapes and introduce practical strategies to make the process easier.

1.0What Are Irregular Shapes?

Irregular shapes are figures that do not have standard geometric properties like symmetry or uniform side lengths. Unlike rectangles, triangles, or circles, irregular shapes do not adhere to predefined formulas, which makes finding the area of irregular shapes less straightforward.

2.0Methods for Finding the Area of Irregular Shapes

Area of Irregular Shapes by Counting Squares

One of the simplest methods for estimating the area of irregular shapes is by counting the number of square units within the shape on a grid.

Step-by-Step Approach:

• Overlay a grid with known square units over the shape.
• Count the full squares completely within the shape.
• Count the partial squares and estimate their combined value (e.g., two half squares can be counted as one full square).
• Sum up the values for an approximate area.

This method is practical when precision is not critical and is commonly used in classrooms for teaching basic surface area of irregular shapes estimation.

Example 1: Find the Area of the given shape.

Solution: 

The area of a given shape is total number of squares. 

So, the area is 15 square units.

Example 2: Find the area of the given Irregular shape.

Solution: 

To find the area of the irregular shapes just calculate the number of squares.

Let's count the squares that are fully covered, half-covered, and so on.

For fully covered squares and More than half covered squares we take 1 square units, for Half covered squares we should assign \frac{1}{2} and for less than half covered squares we assign 0.

Therefore, area of irregular shape:

$\text{ Area }=5(1)+6(1)+12\left(\frac{1}{2}\right)+13(0)$

$\text{ Area }=5+6+6$

$\text{ Area }=17$

Decomposition Method

The decomposition method involves breaking down an irregular shape into smaller, regular shapes, such as triangles, rectangles, or circles. You can then use standard formulas to find the area of each segment and sum them up.

Example:

Identify and divide the shape into known geometrical figures.

Calculate the area of each component using appropriate formulas:

Rectangle: Area = length × width 

Triangle:  $Area=\frac{1}{2}\times base\times height$

Add all the individual areas to find the total area of the irregular shape.

Note: This method works well when the irregular shape can be closely approximated by simpler geometric figures.

Example 1: How can we find the area of Given irregular shape.

Solution: 

Area of Irregular shape = Area of Triangle (1 + 3 + 4 + 6 + 7) + Area of Square 2 + Area of Parallelogram 5

Example 2: Compute the area of Irregular shape 

Solution: 

Area of Irregular shape = Area of Square + Area of Rectangle

$\text{ Area }=4\times \text{ Side }+\text{ Length }\times \text{ Breadth }$

$\text{ Area }=4\times 6+4\times 7$

$\text{ Area }=24+28$

$\text{ Area }=52$

Using Coordinate Geometry

For more precise finding of the area of irregular shapes, especially in cases involving complex polygons, coordinate geometry can be used. This involves plotting the vertices of the shape on a coordinate plane and using the following formula:

$Area=\frac{1}{2}\left|{\sum }_{i=1}^{n-1}(x_i{y}_{i+1}-x_{i+1}{y}_1)+(x_n{y}_1-x_1{y}_n)\right|$

Where $(x_i,y_i)$ are the coordinates of the vertices, and n is the number of vertices.

3.0Practical Applications of Estimating the Area of Irregular Shapes

• Geographical Mapping: Calculating the area of irregular land plots.
• Biology: Determining the leaf area or other natural objects with irregular outlines.
• Architecture: Designing spaces that do not conform to regular geometric shapes.

4.0Tips for Accurately Estimating the Area of Irregular Shapes

• Use tools like graph paper for area of irregular shapes by counting squares.
• For more precision, opt for digital tools or software that can handle complex coordinate geometry or calculus-based solutions.
• Practice with different shapes to get comfortable with the methods discussed.