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What Is the Area of a Polygon? Formula & Examples

Updated: October 13, 2026

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Category: Advance Math, Math

The area of a polygon is the total amount of two-dimensional space enclosed within its straight-line boundaries. Measured in square units (such as square centimeters, square meters, or square inches), this region represents the surface flat space occupied by any closed multi-sided geometric shape.

Whether you are calculating the surface of a simple triangle or a complex 12-sided figure, understanding what is the area of a polygon depends entirely on the shape's symmetry and number of sides.

Direct Answer Summary: Formulas at a Glance

Polygon Type	Primary Condition	Area Formula
Regular Polygon	Equal sides, equal angles, and given apothem (a)	A = ½ × Perimeter × Apothem A = ½Pa
Regular Polygon	Equal sides (s) and number of sides (n), without apothem	A = ns² / [4 tan(π/n)]
Triangle	Base (b) and height (h)	A = ½ × b × h
Rectangle	Length (l) and width (w)	A = l × w
Irregular Polygon	Any simple, non-self-intersecting polygon	Shoelace Formula: A = ½ \|Σ(xᵢyᵢ₊₁ − xᵢ₊₁yᵢ)\|

What is the Area of a Regular Polygon?

When asking what the area of a regular polygon is, the calculation relies on symmetry. A regular polygon has sides of equal length and angles of equal measure (such as an equilateral triangle, a square, or a regular pentagon).

The Apothem Formula

To determine the area of a regular polygon, mathematicians use the apothem—the perpendicular distance from the center of the polygon to the midpoint of any side. This measurement helps calculate the area using the polygon's perimeter.

Area of a Regular Polygon Formula

A = ½ × P × a

Area = ½ × Perimeter × Apothem

What Do the Symbols Mean?

A = Area

The total space enclosed within the regular polygon, measured in square units.

P = Perimeter

The total length of all the sides. For a regular polygon:

P = n × s

a = Apothem

The perpendicular distance from the center of the polygon to the midpoint of one of its sides.

Additional symbols:

n: Number of sides in the polygon.
s: Length of each side.

What is the Formula for Area of a Polygon?

Depending on whether a polygon is regular or irregular, different formulas can be used to calculate its area. If the apothem is unknown or the polygon has unequal sides, the following methods can help determine its area.

1. General Regular Polygon Formula (Using Side Length)

If you know the number of sides (n) and the length of each side (s) but do not know the apothem, you can use trigonometry to calculate the area of a regular polygon.

Formula

A = ns² / [4 tan(180° / n)]

Where:

A: Area of the regular polygon.
n: Number of sides.
s: Length of each side.
tan: Tangent trigonometric function.

2. Irregular Polygon Formula (Shoelace Formula)

If you need to calculate the area of an irregular polygon with unequal side lengths on a Cartesian plane, you can use the Shoelace Formula, also known as Gauss's Area Formula.

First, identify the coordinates of each vertex in order:

(x₁, y₁), (x₂, y₂), …, (xₙ, yₙ)

The area is calculated using the following formula:

A = ½ |S₁ − S₂|

S₁ = x₁y₂ + x₂y₃ + ⋯ + xₙy₁

S₂ = y₁x₂ + y₂x₃ + ⋯ + yₙx₁

Where:

A: Area of the irregular polygon.
(x₁, y₁), …, (xₙ, yₙ): Coordinates of the polygon's vertices, listed in order around its boundary.
| |: Absolute value, which ensures the calculated area is non-negative.

Important: List all vertices in consecutive boundary order and repeat the first vertex at the end when arranging the coordinate pairs for calculation.

Step-by-Step Example: Calculating Polygon Area

Here is a practical example showing how to apply the regular polygon area formula to solve a geometry problem step by step.

Problem

Find the area of a regular hexagon with a side length of 6 cm and an apothem of 5.2 cm.

Side length (s) = 6 cm

Apothem (a) = 5.2 cm

Number of sides (n) = 6

Step-by-Step Solution

Step 1: Calculate the Perimeter (P)

A regular hexagon has six equal sides. Multiply the side length by the number of sides.

P = n × s
P = 6 × 6 cm
P = 36 cm

Step 2: Apply the Area Formula

Use the perimeter and apothem in the formula for the area of a regular polygon.

A = ½ × P × a
A = ½ × 36 × 5.2

Step 3: Calculate the Final Area

First, divide the perimeter by 2, then multiply the result by the apothem.

A = 18 × 5.2
A = 93.6 cm²

Final Answer

Area = 93.6 cm²

The area of the regular hexagon is 93.6 square centimeters.

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Frequently Asked Questions

1. What is the area of a polygon in simple terms?

The area of a polygon is the two-dimensional region covered inside its straight boundary lines, expressed in square units.

2. What is the formula for the area of a polygon with 5 sides (Pentagon)?

For a regular pentagon with side length s, the area formula is:

A = ¼√(5(5 + 2√5)) × s²

Approximately: A = 1.72047 × s²

3. How do you find the area of an irregular polygon without coordinates?

You can divide (triangulate) the irregular polygon into non-overlapping triangles, calculate the individual area of each triangle using base and height or Heron's formula, and sum them together.

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