Area of polygon helps us determine how much space a multi-sided shape covers. Whether you're working with simple shapes like triangles and rectangles or more complex ones like hexagons and irregular quadrilaterals, each polygon has its own method for area calculation.

Polygons are classified as regular (equal sides and angles) or irregular (unequal sides or angles). For regular polygons, formulas are direct and consistent. 

For example, the area of a square is simply the side length squared. In contrast, the area of an irregular polygon is often calculated by breaking it into simpler shapes such as triangles or rectangles, then adding the individual areas together.

In this blog, we will learn to calculate the area of polygon using different formulas.

Area of Polygon Definition

The area of a polygon refers to the total space enclosed within its sides on a two-dimensional (2D) plane. In simpler terms, it is the surface or region covered by a polygon, measured in square units such as cm², m², or in².

If a polygon has n sides, the area is the measure of space bound by all these n sides. The shape could be a triangle, quadrilateral, pentagon, or any multi-sided figure.

There are two major categories of polygons when it comes to calculating area:

• Regular polygons: All sides and angles are equal (e.g., square, equilateral triangle).

• Irregular polygons: Sides and angles are not equal (e.g., a land plot with uneven boundaries).

Why Learn to Find the Area of Polygon?

Understanding how to calculate the area of a polygon is useful in many real-life and academic situations. For example:

• Estimating materials like paint, tiles, or flooring for a space

• Creating virtual objects and environments in game design or computer graphics

• Surveying plots of land for farming or construction

• Solving geometry problems in exams like the SAT, GRE, or school tests

Steps to Find the Area of Polygon

Here are the different you can follow the find the area of formula:

Step 1: Identify the type of polygon: Is it regular or irregular? Are coordinates given?

Step 2: Choose the right formula: Use apothem method for regular polygons, coordinate method for plotted shapes, and triangle division for others.

Step 3: Plug in values carefully: Check units, especially when using side lengths.

Step 4: Verify your answer: Units should be in square units (e.g., cm², m²).

Step 4: Apply real-world logic: Ask if the number makes sense in the context (e.g., floor plan, land size).

Area of Polygons Formulas

Area of a polygon can be calculated using different formulas depending on whether the shape is regular or irregular. Below are the standard formulas for calculating the area of common polygons often encountered in daily life:

• Area of a Triangle = (1/2) × base × height

• Area of a Rectangle = length × width

• Area of a Parallelogram = base × height

• Area of a Trapezium = (1/2) × (sum of the lengths of the parallel sides) × height

• Area of a Rhombus = (1/2) × (product of the diagonals)

• Area of a Quadrilateral = (1/2) × diagonal × (sum of perpendicular heights from opposite vertices)

Read More: Acute Angle

Area of Regular Polygon

A regular polygon has all sides and angles equal. The area depends on the shape, and different polygons use different formulas. Below are some commonly used are of regular polygon formulas:

• Equilateral Traingle:  (√3 / 4) × side²

• Square: Side × Side

• Regular pentagon: (5/2) × side × apothem

For any regular polygon, you can also use the general formula:

Area = (½) × Perimeter × Apothem
Or
Area = (½) × n × side × Apothem

 Where n is the number of sides.

Example:  If the perimeter is 20 cm and the apothem is 5 cm:

 Area = (½) × 20 × 5 = 50 cm²

Area of Irregular Polygon

An irregular polygon has unequal sides and angles. To find its area, divide it into simpler shapes (like triangles or quadrilaterals), calculate their areas, then add them together.

Example: An irregular quadrilateral ABCD is divided into two triangles by diagonal AC.  Let:

AC = 10 cm

Height from B to AC (h₁) = 4 cm

Height from D to AC (h₂) = 5 cm

Area = (½) × AC × (h₁ + h₂)
= (½) × 10 × (4 + 5) 

= (½) × 10 × 9 = 45 cm²

Read More: Measurement

Area of Polygon Solved Examples

1. Find the area of a triangle with a base of 10 cm and a height of 6 cm.

Solution:
We use the formula:

Area = ½ × base × height

Area= ½ × 10 × 6 

5 × 6 = 30 cm2

 Answer: 30 cm²

2. A rectangle has a length of 15 meters and a width of 8 meters. What is its area?

Solution: 

Use the formula:

Area = length × width

Area = 15 × 8 =120 m²

Answer: 120 m²

3. Find the area of a parallelogram with a base of 14 cm and a height of 9 cm.

Solution:
Use the formula:

Area = base × height

Area=14 × 9 =126 cm²

Answer: 126 cm²

4. A trapezium has two parallel sides measuring 12 cm and 7 cm, and the height between them is 5 cm. Find the area.

Solution:

Use the formula:

Area = ½ × (a + b) × height

Where a = 12 cm, b = 7 cm, height = 5 cm

Area = 12 × (12 + 7) × 5 =12 × 19 × 5 = 9.5 × 5 = 47.5 cm²

Answer: 47.5 cm²

5. A rhombus has diagonals of length 18 cm and 10 cm. What is its area?

Solution:
Use the formula:

Area =½ × d1 × d2

Where d1=18 and d2 = 10 cm

Area =12 × 18 × 10 = 9 × 10 = 90 cm²

Answer: 90 cm²

6. In a quadrilateral, a diagonal measures 16 m. The perpendicular distances from the two opposite vertices to this diagonal are 4 m and 6 m. Find the area of the quadrilateral.

Solution:
Use the formula:

Area = ½ × Diagonal × (height1+height2)

Area = ½ × 16 × (4 + 6) = 8×10 = 80 m²

Answer: 80 m²

Did You Know?

• The word "polygon" comes from Greek: poly (many) + gon (angles).

• A shape with more than 12 sides is often just called an "n-gon".

• The formula for the area of polygons becomes especially useful in computer vision and robotics.

Also Read: Find the Angle of a Triangle

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