Summarize this article:
102 LearnersLast updated on September 15, 2025
Area of Polygon
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.
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.
Mistake 1
Using the wrong formula for the type of polygon
Ensure the correct formula is used based on whether the polygon is regular or irregular.
Regular polygons have a unique formula compared to irregular ones.
Mistake 2
Confusing the apothem with the radius
In regular polygons, the apothem is not the same as the radius.
The apothem is the perpendicular distance from the center to a side.
Mistake 3
Incorrectly applying the Shoelace formula
Make sure to follow the order of vertices when using the Shoelace formula.
Incorrect ordering can lead to errors in calculations.
Mistake 4
Neglecting to convert units
Ensure all measurements are in the same unit before calculating the area.
Convert them if necessary to avoid discrepancies.
Mistake 5
Forgetting to divide by two in the Shoelace formula
The Shoelace formula requires dividing by two to get the correct area.
Forgetting this step can double the result incorrectly.
Area of Polygon Examples
Problem 1
The perimeter of a regular hexagon is 60 cm, and its apothem is 5 cm. What will be the area?
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
Problem 2
Find the area of a polygon with vertices at (1,1), (4,1), (4,5), and (1,5).
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.
Problem 3
A regular pentagon has a side length of 8 m and an apothem of 5.5 m. What is the area?
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
Problem 4
Calculate the area of a triangle with vertices at (2,3), (6,7), and (8,3).
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.
Problem 5
Help Sarah find the area of a regular octagon with a side length of 10 cm and an apothem of 12 cm.
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
FAQs on Area of Polygons
1.Can the area of a polygon be negative?
2.How to find the area of a regular polygon if the side length and apothem are given?
3.How to find the area of an irregular polygon using coordinates?
4.How is the perimeter of a regular polygon calculated?
5.What is meant by the area of a polygon?
Glossary of Terms Related to Area of Polygons
- Polygon: A closed, two-dimensional shape with straight sides.
- Apothem: The distance from the center of a regular polygon to the midpoint of one of its sides.
- Shoelace Formula: A method to calculate the area of a polygon when the vertices are known.
- Perimeter: The total length of all sides of a polygon.
- Regular Polygon: A polygon with all sides and angles equal.
Explore More measurement
![Important Math Links Icon]()
Previous to Area of Polygon
![Important Math Links Icon]()
Next to Area of Polygon
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
