# hexagon

In this section, we will introduce you to the hexagon formula.

You will also learn about hexagon shape, hexagon examples, hexagonal shapes, hexagon dimensions, irregular hexagon, as well as the area of a hexagon using real-life examples.

Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions on Hexagons at the end of the page.

As the name suggests, a hexagon is made up of six sides.

The name is divided into hex, which means six, and gonia, which means corners.

A hexagon is a two-dimensional shape as it has only length and width.

## Lesson Plan

 1 What is a Hexagon? 2 Solved Examples on Hexagon 3 Interactive Questions on Hexagon

## What is a Hexagon?

Hexagonal shape definition is given as a two-dimensional geometrical shape which is made of six sides, having the same or different dimensions of length.

Let’s find some real-life hexagon examples.

Have you ever observed a pencil? Now, let’s turn it around and see how it looks?

Does that shape resemble your pencil too? Let’s count the number of sides it has.

Let’s see one more example:

Can you spot the above shape in the image below?

Yes, the sofa resembles the green figure above. Let's count the number of sides too.

Both in Shape 1 and Shape 2, the number of sides is 6, isn't it?

Hence both are hexagonal in shape as the number of sides is 6. Both the hexagon shapes are characterized by a difference in their appearance and dimensions.

## Types of Hexagonal shape

Think for a bit, just observing the sides. Do the two figures below look similar? Let’s compare them.

The first shape, set in a form where the measurement of all sides of the hexagon is similar, is called a regular hexagon.

The second shape, set in a form where the measurement of each side of the hexagon is dissimilar, is called an irregular hexagon.

To understand the concept of regular and irregular hexagons better, let's have a look at the simulation below.

### More About the Hexagon Shape and Its Possible Dimensions

Hexagons can further be classified into two categories based on the measurement of angles formed by sides of hexagons.

Convex Hexagon

Concave Hexagon

When each internal angle in a hexagon measures less than 180˚, it is called a convex hexagon.

When one or more internal angles in a hexagon measure more than 180˚, it is called a concave hexagon.

## Properties of Hexagon

Properties of Hexagon are as follows:

1.  A hexagon has six sides.

2.  It also has six angles.

3.  It has six corners.

Important Notes
• A regular hexagon is also a convex hexagon because the internal angles are less than 180 degrees.

## Unique Properties of a Regular Hexagon

The uniqueness in this hexagonal shape is explained by the below mentioned regular hexagon properties and equations.

1. The length of each side of a regular hexagon is equal.

2. It is a symmetrical shape since each side is of equal length.

3. Opposite sides of a regular hexagon will always be parallel to each other.

4. A regular hexagon can be split into 6 equilateral triangles.

5. Each angle of a regular hexagon is equal and measures 120˚

1. Formula for number of diagonals is given as per the equation below where n=6 for a regular hexagon. Hence it has 9 diagonals.
 $$\dfrac{n(n-3)}{2}$$

\begin{align} \text{Number of diagonals}&=\dfrac{6(6-3)}{2} \\ &=\dfrac{6 \times 3}{2} \\ &=\dfrac{18}{2}\\ \therefore\ \text{Number of diagonals}&= 9 \end{align}

1. From these 9 diagonals, 6 lines pass from the center point. Let’s take a close look and count the number of lines passing from center too.

1. The Sum of internal angles formed by a regular hexagon is 720˚ (because each angle is 120˚ and there are 6 such angles adding up to 720˚).

It is given by the formula for regular polygon, where n is number of sides, which has value 6 for hexagonal shape;
 $$\text{Sum of interior angles}=\!(n-2)\times180˚$$

\begin{align} &\!\!=\!\!(6-2)\times180˚\\ &=4\times180˚\\ \therefore \text{Sum of interior angles}&=720˚ \end{align}

Think Tank
• Hexagon falls in a category of shapes that can fill a plane without wasting it, just like the figure below. Can you think of the other two shapes, which fall under the same category?

## Area of Regular Hexagons

Let’s say, each side of a regular hexagon is named s. To find the area of a hexagon we use the following formula.

This formula of the area is for hexagons which are regular hexagons only. The hexagon dimensions need to be the same for its applicability.

## Solved Examples

 Example 1

Rosie saw two real-life hexagon examples which are very easily found. Give two real-life hexagonal shape examples.

Solution

 $$\therefore$$ Nut & Honeycomb are 2 examples
 Example 2

Max drew two kinds of hexagonal shapes.

A. First hexagon's dimensions were all sides being 2 cm

B. Second hexagon's dimensions were all sides having variable lengths as below.

• Side 1 = 2 cm
• Side 2 = 3 cm
• Side 3 = 3 cm
• Side 4 = 2 cm
• Side 5 = 4 cm
• Side 6 = 4 cm

Which among the two hexagons drawn by Max would be a regular hexagon and which one will be an irregular hexagon?

Solution

A is a regular hexagon as all sides have the same length.

B is an irregular hexagon as all sides have different lengths, which indicates the irregularity in the dimensions.

 $$\therefore$$ A - Regular Hexagon, B - Irregular Hexagon
 Example 3

Rita has to find the area of a regular hexagon. If the value of one side is 3 cm, what would be the hexagon area?

Solution

Applying the formula of area of regular hexagon,

$\text{A} =\dfrac{(3\sqrt{3} \text s^2)}{2}$

$$\text{Now, as s = 3 cm}$$

\begin{align}\text{A}&=\dfrac{(3\sqrt{3}\times 3^2)}{2}\\ \text{A}&= 23.382 \text{ cm}^2 \end{align}

 $$\therefore$$ Area of regular hexagon = 23.382 cm2
 Example 4

Katie was asked to find out the length of each side of a regular hexagon. If the hexagon area is 100 cm2, what do you think the length of each side of the hexagon will be?

Solution

Applying the formula of area of regular hexagon,

\begin{align} \text{A}&=\dfrac{(3\sqrt{3}s^2)}{2}\\\\ \text{Assume s = x cm}\\\\ \dfrac{(3\sqrt{3}\times \text x^2)}{2}&=100 \text{ cm}^2\\ \text x&= 6.204 \text{ cm} \end{align}

 $$\therefore$$ Length of side = 6.204 cm.
 Example 5

Leo was asked to prove that a regular hexagon when split into 6 parts, forms 6 equilateral triangles. How do you think he could have proved it?

Solution

He gave the following reasoning to prove his point:

As the value of each internal angle is 120˚, the line joining 2 opposite points in hexagon will divide the internal angle into half, forming 60˚

This will happen simultaneously with all the sides which will leave us with only one angle of the hexagon unknown.

Hence applying the sum of 3 angles of a triangle is 180˚, we obtain 3rd angle to be 60˚

This proves the regular hexagon, when split into 6 parts, forms 6 equilateral triangles.

## Interactive Questions

Here are a few activities for you to practice. Select/Type your answer and click the "Check Answer" button to see the result.

## Let's Summarize

We hope you enjoyed learning about hexagons with the simulations and interactive questions. Now you will be able to easily solve problems on the hexagon shape definition, hexagon formula, hexagon examples, hexagon shapes, hexagon dimensions, hexagon sides, properties of hexagon, irregular hexagon, and the area of a hexagon along with the real-life examples.

At Cuemath, our team of math experts is dedicated to making learning fun for our favorite readers, the students!

Through an interactive and engaging learning-teaching-learning approach, the teachers explore all angles of a topic.

Be it worksheets, online classes, doubt sessions, or any other form of relation, it’s the logical thinking and smart learning approach that we at Cuemath believe in.

## 1. What is a hexagon?

Hexagon shape definition is given as a two-dimensional geometrical shape of 6 sides.

## 2. Are all 6-sided shapes hexagons?

Yes, all 6-sided shapes are hexagons.

## 3. What are the three attributes of a hexagon?

The three attributes of a hexagon are:

• It has 6 sides
• It has 6 angles
• It has 6 corners

## 4. Does a hexagon have equal sides?

Hexagon may not necessarily have all sides equal. It can have sides with variable lengths too.

The hexagon having equal sides is called a regular hexagon and the one with variable sides is called an irregular hexagon.

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