The name hexagon consists of the two Greek words which are ‘Hexa’ which means six and ‘Gonia’ which means corners. Hexagon in geometry is a polygon with six sides and six angles. A hexagon is a two-dimensional closed shape which has six sides. The space that is enclosed by all its sides is known as the area of a hexagon.

In this math article, we will learn about the area of a hexagon and the formula used to find the area of a hexagon, the type of Hexagonal, the area of the hexagon with apothem, and the area of a hexagon with radius.

What is a Hexagon

In geometry, a hexagon can be defined as a closed two-dimensional polygon with six sides. Hexagon has 6 vertices and 6 angles also.

A hexagon is made of straight lines and does not have any curved sides. The Hexagon shape must also be closed so all the sides touch each other. The blow images represent figures that are not hexagons

Types of Hexagon

The hexagonal shape is classified into several types based on the measure of sides and angles.

Regular Hexagon

When the length of all the sides and the measure of all the angles are equal, it is a regular hexagon. The value of each interior angle of a regular hexagon is 120 degrees. The diagonals of regular hexagons are equal and the diagonals intersect at the center of the hexagon.

The above image is for a regular hexagon.

 Irregular Hexagon

An irregular hexagon is a hexagon which has different measurements of side lengths and angles. The result is a shape which is different and hardly resembles a regular hexagon. In an irregular hexagon, all the internal angles are not equal to 120° but the sum of the internal angles is equal to 720°.

The above image is for an irregular hexagon.

What is the Area of a Hexagon?

The area of a hexagon is the total amount of space which is enclosed by all six sides. A hexagon has a 6-sided, 2-dimensional geometric figure. The sum of a hexagon’s internal angles is 720°. A regular hexagon consists of 6 rotational symmetries and 6 reflection symmetries. The value of each internal angle of a regular hexagon is 120 degrees. Because the hexagon is a two-dimensional shape, its area also lies in a two-dimensional plane. The unit of area of the hexagon is given in square units such as sq. m, sq. cm, sq. in, or sq. ft.

The image below shows the area of a regular hexagon with the help of a hexagon placed on a graph paper.

Area of a Hexagon Formula

The area of the hexagon is the space confined within the sides of the polygon.

The area of a hexagon can be represented as follows.

Area of a Hexagon = \( \frac{3\sqrt{3}}{2}s^{2} \)

Where, “s” is a side length of the hexagon.

How to Find the Area of a Hexagon?

The area of a hexagon is calculated if the side length of the hexagon is given. The formula for area of regular hexagon =\( \frac{3\sqrt{3}}{2}s^{2} \)

The following steps are followed to find the area of the hexagon:

Step 1: Identify the regular hexagon side’s length

Step 2: Find the area of the hexagonal using the formula, Area of hexagon = \( \frac{3\sqrt{3}}{2}s^{2} \); where ‘s’ is the side length of the hexagonal.

Step 3: Denotes the final answer in square units.

Example: Find the area of a regular hexagon which has a side length of 10 inches.

Solution: Given the length of the side of a regular hexagon,s = 10 inches.

Area of a regular hexagon =\( \frac{3\sqrt{3}}{2}s^{2} \)

=\( \frac{3\sqrt{3}}{2}\times 10^{2} \)

= \( 150\sqrt{3} \)

= 259.80 in2

Answer: The area of the regular hexagon is 259.80 in2.

Learn about Area of Pentagon

Area of Hexagon with Apothem

The apothem of a hexagon is a line segment that is drawn from the center and perpendicular to the sides of the hexagon. So, the area of the hexagon with apothem = 1/2 x perimeter of the hexagon x apothem.

Area of hexagon with apothem =\( \frac{1}{2}\times n\times m [/latex

Applying formula \(\) m=6s \)

Area of hexagon with apothem = \( \frac{1}{2}\times n\times 6\times s \)

=\( 3\times n\times s \)

= \( 3 n s \)

Where,

‘n’ = the length of an apothem

‘m’ = the perimeter of the hexagon 

‘s’ = the length of the hexagon’s side.

In the above figure, the formula for the area of a regular hexagon with the apothem can be observed.

Area of a Hexagon with radius

The area of the hexagon is considered as the area enclosed by its six sides. The area of a regular hexagon is represented as;

Area of a regular hexagon = \( \frac{3\sqrt{3}}{2}s^{2} \)

In a regular hexagon, the radius (r) of a hexagon is equal to the length of its side (s). It means s = r.

Therefore, the regular hexagon can be divided into six equilateral triangles.

The area of a regular hexagon with radius “r” = \( \frac{3\sqrt{3}}{2} r^{2} \).

Area of Irregular Hexagon

There is no direct formula to find the area of an irregular hexagon. However we can use some indirect and graphical methods to find its area. Let’s learn how to find the area of an irregular hexagon.

Area of irregular hexagon from vertices

The following steps are followed to find the area of irregular hexagon ABCDEF from vertices as mentioned in below figure:

 Step 1: List the x and y coordinates of all vertices of the irregular hexagonal. First we need to create a chart with two columns and seven rows. Each row is labeled with the names of the six points (point A, point B, point C, etc.), and each column is labeled with the x or y coordinates of those points. We write down the x and y coordinates of point A to the right of point A, the x and y coordinates of point B to the right of point B, and so on. Repeat the coordinates of the first point at the end of the list.

The points coordinates, in (x, y) format are as follows.

• A: (4, 10)
• B: (9, 7)
• C: (11, 2)
• D: (2, 2)
• E: (1, 5)
• F: (4, 7)
• A (again): (4, 10)

Refer to the above figure 1.

Step 2: Multiplies the x-coordinate of each point by the y-coordinate of the next point.

We can think of this as drawing a diagonal line to the right and one line down from each x-coordinate. List the results to the right of the table. Then add the results.

• \( 4 \times 7 = 28 \)
• \( 9 \times 2 = 18 \)
• \( 11 \times 2 = 22 \)
• \( 2 \times 5 = 10 \)
• \( 1 \times 7 = 7 \)
• \( 4\times 10 = 40 \)

\( 28 + 18 + 22 + 10 + 7 + 40 = 125 \)

Refer to the below figure.

Step 3: Multiplies the y-coordinates of each point by the x-coordinates of the next point.

Imagine drawing a diagonal line from each y-coordinate down and left to the x-coordinate below. Once we have multiplied all of those coordinates together, add the results together.

• \(10 \times 9 = 90 \)
• \( 7\times 11 = 77 \)
• \( 2 \times 2 = 4 \)
• \( 2 \times 1 = 2 \)
• \( 5 \times 4 = 20 \)
• \( 7 \times 4 = 28 \)

\( 90 + 77 + 4 + 2 + 20 + 28 = 221 \)

Refer to the below figure.

Step 4: Subtracts the sum of the second set of coordinates from the sum of the first set of coordinates.

Subtracting 221 from 125. \(125 – 221 = -96 \).Now considering the absolute value of this answer: 96. The area can only be positive.

Refer to the below figure.

Step 5: Divide this difference by two

Dividing 96 by 2 and we get the area of the irregular hexagon. \( \frac{96}{2} =48 \)

Therefore, the area of the irregular hexagon is 48 square units.

Refer to the below figure.

Area of irregular hexagon from shapes

Irregular hexagons have sides that are not equal in length. There is no single equation to calculate the area of an irregular hexagon. The easiest way to calculate the area of the irregular hexagonal is to divide it into calculable shapes and add the total areas of those shapes. In this case, several dimensions are needed to determine the total area of the Irregular hexagons.

Important Points on Area of Hexagon

• The area of a hexagon is the total amount of space that is enclosed by all six sides. A hexagon has a 6-sided, 2-dimensional geometric figure.
• The area of hexagon is represented by \( \frac{3\sqrt{3}}{2}s^{2} \)

where “s” denotes the sides of the hexagon.

• Based on the Classification of hexagons based on their sides, there are two types of hexagons which are regular hexagons and irregular hexagons.
• The Area of a hexagon with radius “r” is \( \frac{3\sqrt{3}}{2} r^{2} \).
• The area of the hexagon with apothem is \( 3ns \).

Where,’n’ = the length of an apothem and ‘s’ = the length of the side of the hexagon.

Area of Hexagon Solved Examples

Example 1. What is the area of a regular hexagon inscribed in a circle of radius p.

A1. A regular hexagon can be divided into 6 equilateral triangles.

Since the hexagon is inside a circle of the radius p is one side of an equilateral triangle.

So the area of a hexagon = \( 6\times \frac{\sqrt{3}}{4}p^{2} \)

= \(\frac{3\sqrt{3}}{2}p^{2} \) square units

The area of a regular hexagon inscribed in a circle of radius p is \(\frac{3\sqrt{3}}{2}p^{2} \) square units.

Example 2: What is the area of a regular hexagon whose side length is equal to 6 cm?

A2. Given, the side length of the hexagon, s= 6 cm.

By the formula of hexagon, we know that;

The area of hexagon =\( \frac{3\sqrt{3}}{2} s^{2} \)

The area of hexagon =\( \frac{3\sqrt{3}}{2} \times 6^{2} \)

The area of hexagon = \( 93.53 sq.cm \)

Answer: the area of a regular hexagon is  93.53 sq.cm.

Example 3: If the apothem of a regular hexagon is 5 cm and the side length is 4 cm, then find the area of the hexagon.

A3. Given, apothem of hexagon is, n = 5 cm

Side length is, s = 4 cm

By the formula, we know that;

Area of hexagon = \( 3ns \)

Area of hexagon =  \( 3 x 5 x 4 \)

= \( 60 sq.cm \).

Answer: the area of a regular hexagon is 60 sq.cm.

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