Area of Polygon
2026-02-28 13:01 Diff
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Last updated on September 15, 2025

Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of a polygon.

What is the Area of a Polygon?

A polygon is a two-dimensional figure with three or more straight sides. The area of a polygon is the total space it encloses.

The most common types of polygons include triangles, rectangles, and pentagons, each with specific formulas to calculate their area based on their dimensions and properties.

Area of a Polygon Formula

The formula for finding the area of a polygon depends on the type of polygon. For example, the area A of a regular polygon can be calculated using the formula: A = (1/2) × Perimeter × Apothem, where the apothem is the distance from the center to the midpoint of a side.

For irregular polygons, the area can be determined by dividing the polygon into simpler shapes, such as triangles, and summing their areas.

How to Find the Area of a Polygon?

We can find the area of a polygon using various methods depending on the information available. They are: Method for Regular Polygons For regular polygons, use the formula A = (1/2) × Perimeter × Apothem.

Method for Irregular Polygons Divide the polygon into triangles and calculate the area of each triangle, then sum them up.

Method Using Coordinates For a polygon with vertices given as coordinates, use the Shoelace formula: A = (1/2) × |Σ(x_i*y_(i+1) - y_i*x_(i+1))|, where the sum is over all the vertices and the last vertex is connected back to the first.

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Unit of Area of Polygon

We measure the area of a polygon in square units.

The measurement depends on the system used:

In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

Special Cases or Variations for the Area of Polygons

Depending on the type of polygon and the given dimensions, different methods are used to calculate the area. Here are some special cases:

Case 1: Regular Polygons If the polygon is regular (all sides and angles are equal), use the formula A = (1/2) × Perimeter × Apothem.

Case 2: Triangular Decomposition For irregular polygons, divide the shape into triangles and sum their areas.

Case 3: Coordinate Method For polygons with vertices given as coordinates, apply the Shoelace formula for a straightforward calculation.

Tips and Tricks for Area of Polygons

To ensure accurate results when calculating the area of polygons, consider the following tips and tricks:

  • Always verify the type of polygon to select the appropriate formula.
     
  • For regular polygons, ensure accurate measurement of the apothem.
     
  • Use the Shoelace formula for polygons with coordinate data to avoid manual errors.

Common Mistakes and How to Avoid Them in Area of Polygons

It is common for students to make mistakes while finding the area of polygons. Let’s take a look at some mistakes made by students.

Problem 1

The perimeter of a regular hexagon is 60 cm, and its apothem is 5 cm. What will be the area?

Okay, lets begin

We will find the area as 150 cm²

Explanation

For a regular hexagon, use the formula:

A = (1/2) × Perimeter × Apothem.

A = (1/2) × 60 × 5 = 150 cm²

Well explained 👍

Problem 2

Find the area of a polygon with vertices at (1,1), (4,1), (4,5), and (1,5).

Okay, lets begin

The area is 12 square units.

Explanation

Using the Shoelace formula for the vertices (1,1), (4,1), (4,5), (1,5), we compute:

A = (1/2) × |(1*1 + 4*5 + 4*5 + 1*1) - (1*4 + 1*4 + 5*1 + 5*1)| = 12 square units.

Well explained 👍

Problem 3

A regular pentagon has a side length of 8 m and an apothem of 5.5 m. What is the area?

Okay, lets begin

We find the area as 110 m²

Explanation

For a regular pentagon, use the formula:

A = (1/2) × Perimeter × Apothem.

Perimeter = 5 × 8 = 40 m.

A = (1/2) × 40 × 5.5 = 110 m²

Well explained 👍

Problem 4

Calculate the area of a triangle with vertices at (2,3), (6,7), and (8,3).

Okay, lets begin

The area is 12 square units.

Explanation

Using the Shoelace formula for vertices (2,3), (6,7), and (8,3), we have:

A = (1/2) × |(2*7 + 6*3 + 8*3) - (3*6 + 7*8 + 3*2)| = 12 square units.

Well explained 👍

Problem 5

Help Sarah find the area of a regular octagon with a side length of 10 cm and an apothem of 12 cm.

Okay, lets begin

The area is 480 cm²

Explanation

For a regular octagon, use the formula:

A = (1/2) × Perimeter × Apothem.

Perimeter = 8 × 10 = 80 cm.

A = (1/2) × 80 × 12 = 480 cm²

Well explained 👍

FAQs on Area of Polygons

1.Can the area of a polygon be negative?

No, the area of a polygon can never be negative. The area of any shape will always be positive.

2.How to find the area of a regular polygon if the side length and apothem are given?

If the side length and apothem of a regular polygon are given, use the formula A = (1/2) × Perimeter × Apothem.

3.How to find the area of an irregular polygon using coordinates?

For an irregular polygon with coordinates, use the Shoelace formula: A = (1/2) × |Σ(x_i*y_(i+1) - y_i*x_(i+1))|.

4.How is the perimeter of a regular polygon calculated?

The perimeter of a regular polygon is calculated by multiplying the number of sides by the length of one side.

5.What is meant by the area of a polygon?

The area of a polygon is the total space enclosed within its sides.

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables