✅ Hexagon Formula ⭐️⭐️⭐️⭐️⭐

Hexagon Area Formula

In Euclidean geometry, the polygon is a 2D – two-dimensional closed shape that has many sides. Each side has a straight segment. The Hexagon is the type of polygon that consists of 6 sides and angles. The Regular Hexagon consists of six sides and angles which are congruent and are made up using six equilateral triangles. The formula for finding the area of a Regular Hexagon is as follows: Area = (3√3 s2)/ 2 and here S is used for representing the length of the side of the Regular Hexagon.

In Euclidean geometry, a polygon is a two-dimensional closed shape having many sides. Each side is a straight segment. A hexagon is a kind of polygon which has six sides and angles. A Regular hexagon has six sides and angles that are congruent and is made up of six equilateral triangles. The formula to find out the area of a regular hexagon is as given;  Area = (3√3 s2)/ 2 where, ‘s’ represent the length of a side of the regular hexagon.

Methods of Calculating the Area of the Hexagon

There are different ways and methods of calculating the area of the Hexagon, regardless of if you are working with the, or Regular Hexagon. Here we will have a brief look at one of the methods in which you can find the area of the Hexagon.

Method 1: Calculating using the given side length

In this method, first, you need to note the formula for the area of a Regular Hexagon and then input the length of the side that we already know. If you do not know the length of the side then you can find it by following this stepwise approach.

1. Calculate the length of the side.

If you only have the perimeter, then you must divide it by 6 to obtain the length of one of the sides. If you only know apothem, you can still find out the length of sides by using apothem in the formula a = x √3 and then you must multiply the result with 2. This is because the apothem shows x√3 sides of the 30-60-90 triangle which it forms. 

2. Put the value of side length in the formula.

From the mentioned length of one side of the triangle is 8, and then you can just put 8 in the original formula, such as: Area = (3√3 x 8²)/2. Here we get the 3√3 x 64/2. 

Now, 192√3/2 = 166.27. Here it is worth noting that the value of √3 is about 1.732.

In method 2 you can calculate with the given apothem. In the 1st step of this method, the formula is used for finding the area of the Hexagon with the given apothem. In the next step, the apothem is used for finding the perimeter. And finally, in the last step, all the known values are plugged into the formula.

Solved Examples of the Hexagon Formula

Vedantu is a reliable platform for online learning resources and notes on a wide range of subjects and topics. At Vedantu you can find accurate and reliable solved examples of the Hexagon that can help you understand the concept even better and give you the required practice needed for the preparation of the exams. For solved examples on Hexagon Formula, you can refer to the Vedantu note here. 

How to Calculate Area of Regular Hexagon Formula

There are different ways to calculate the area of a hexagon, whether you’re working with a regular hexagon or an irregular hexagon. Check below to find the number of ways with which you can easily find the area of the hexagon.

Method 1: Calculate with a Given Side Length

Firstly, write down the area of the regular hexagon formula and insert the length of a side you already know. However, if you are not aware of the length of a side but are given the length of the perimeter or apothem (the height of one of the equilateral triangles created by the hexagon, Which is perpendicular to the side), you can still calculate the length of the side of the hexagon. Here’s how to do it step-by-step:

1. Calculate the Length of a Side:

If you are only given the perimeter, just divide it by 6 to obtain the length of one side. As an example, if the length of the perimeter is 48 cm, then divide it by 6, you get 8cm, the length of the side.

If you only know the apothem, you can still find the length of a side by plugging the apothem into the formula a = x√3 and then multiplying the outcome by 2. It is because the apothem depicts the x√3 sides of the 30-60-90 triangle that it forms. If the apothem is 15√3, for example, then x is 15 and the length of a side is 15 × 2, or 30.

2. Plug the Value of the Side Length into the Formula:

From the aforementioned length of one side of the triangle are 8, just plug 8 into the original formula, like this: Area = (3√3 x 8²)/2. We get 3√3 x 64/2

Then, 192√3/2 = 166.27

Note that the value of √3 is approximately 1.732

Method2: Calculate with a Given Apothem

1. Use the Formula to Find the Area of a Hexagon with a Given Apothem:

The perimeter of the hexagon formula is simply: Area = 1/2 x perimeter x apothem. Let’s say the apothem is 7√3 cm.

2. Use the Apothem to Find the Perimeter

You already know that the apothem is perpendicular to the side of the hexagon; it forms one side of a 30-60-90 triangle. The sides of a 30-60-90 triangle are in the proportion of x-x√3-2x, where:-

• ‘X’ represents the length of the short leg of the triangle, which is through the 30° angle,
• x√3 represents the length of the long leg of the triangle, which is through the 60° angle, and
• 2x represents the hypotenuse of the triangle 2x

The apothem is the side denoted by x√3. Thus, we need to plug the length of the apothem into the formula a = x√3 and solve. As an example, if the apothem’s length is 7√3, plug it into the formula and obtain 7√3 cm = x√3, or x = 7 cm.

By simplifying for x, you have found the length of the short leg, 7. Since it depicts half the length of one side of the hexagon, multiply it by 2 to get the full length of the side i.e. 7 x 2 = 14 cm.

Since you know that the length of one side is 14cm, multiply it by 6 to find the perimeter of the hexagon: 14 cm x 6 = 84 cm

3. Plug All of the Known Values into the Formula

Now, all you need to do is plug the perimeter and apothem into the formula and solve:

Area = 1/2 x perimeter x apothem

Area = 1/2 x (84) x (7√3) cm

= 509.20 cm²

Solved Examples of Hexagon Formula

Example1:

Find the area of the hexagon with given variables s = 7

Solution1:

we know that the side length of a regular hexagon i.e. s=7

Now, we can find the area of the hexagon using the following formula:

(3√3 s²)/ 2

Let’s substitute the given value into the area formula for a regular hexagon and solve.

A= 3√3 7² / 2

Simplify.

= 3√3 49 / 2

= 147 √3 / 2

= 127.30

Round off the outcome to the nearest whole number.

A=127 cm²

FAQs (Frequently Asked Questions)

What Is Hexagon Formula?

The hexagon formula is defined as a set of formulas to calculate the perimeter, area, and diagonals of the hexagon. The hexagon formulas apply directly to regular hexagons.

Where,

• s = side length.

Perimeter of hexagon: 6s

Where,

• s = side length.

Diagonals of a hexagon: 2s and √3s

Where,

• s = side length

Special Formula: Area = 1/2×perimeter×apothem

Let us see the applications of the hexagon formula in the following solved examples.

Derivation of Hexagon Formula

Consider a regular hexagon of side s. Then,

Area of a hexagon

• Divide the hexagon into small six isosceles triangles.
• Find the area of one of the triangles.
• Multiply by 6.

Perimeter of a hexagon

• The perimeter of any polygon is the sum of all the side lengths.
• A regular hexagon has 6 sides. Thus, multiply the side length by 6.

Thus, the formula for the perimeter of a regular hexagon is given as, P=6×a

Diagonal of a hexagon

• In a hexagon, there are 6(6-3)/2 = 9 diagonals
• These nine diagonals form into six equilateral triangles.
• For longer diagonal, d = 2s, and for shorter diagonal, d = √3s, where s refers to the side of the hexagon.

Thus, the formula for the diagonal of a hexagon is given as, d = 2s, and √3s

Solved Examples Using Hexagon Formula

Example 1: Calculate the perimeter and area of a regular hexagon having a side equal to 4 units.

Solution:

To Find: Perimeter and Area 
Given: s= 4 units.
Using the hexagon formula for perimeter
Perimeter(P) = 6s
P = 6×4
P = 24 units
Using the regular hexagon formula for Area

= 41.56 units²
Answer: Perimeter and area of the hexagon are 24 units and 41.56 units².

Example 2: A hexagonal board has a perimeter equal to 12 inches. Find its area.

Solution:

To Find: Area of the hexagon.
Given: Perimeter = 12  inches.
The perimeter of hexagon = 6s
12 = 6 s 
s = 2 inches.
Using the hexagon formula for Area,

Answer: The area of the hexagonal board is 10.39  inches².

Example 3: Determine the side length of the regular hexagon whose perimeter is 24 units.

Solution:

To Find: Side length of hexagon
Given: perimeter= 24 units.
Using the hexagon formula for perimeter
Perimeter(P) = 6s
24 = 6×s
s = 24/6 units
= 4 units
Answer: The side length of the hexagon is 4 units.

FAQs on Hexagon Formula

What Is Hexagon Formula in Geometry?

The hexagon formula calculates the perimeter, area, and diagonals of the hexagon. The hexagon formulas apply directly to regular hexagons. The hexagon formulas are given

What Is Hexagon Formula For Irregular Hexagon?

In the case of an irregular hexagon, divide the given hexagon into rectangles and right triangles and then find the area of each shape whereas to calculate perimeter, just add the lengths of all its sides.

What is Regular Hexagon?

A regular hexagon is a kind of polygon with 6 equal sides. Its properties are:

• It is having six sides and six angles.
• Lengths of all the sides are equal.
• Measurements of all the angles are equal.
• The total number of diagonals in it is 9.
• The sum of all interior angles will be 720 degrees. Also, each interior angle will be 120 degrees.
• The sum of all exterior angles will be 360 degrees. Also, each exterior angle will be 60 degrees.
• The area of a hexagon will be the region occupied inside its boundary.
• The perimeter of it will be the total length of all 6 sides of it.
• The shape has nine diagonals, means the lines between the interior angles.

Formulas for Hexagon

1] Area of a Hexagon

To find out the area of a hexagon, we have to divide it into small six isosceles triangles. Then, we will find the area of one of the triangles and finally, we will multiply by 6 to find the total area of the polygon.

2] Permiter of a Hexagon

Perimeter will be the sum of all 6 side lengths. The formula for the perimeter of a hexagon is given as:

P=6×a

3] Diagonal of a Hexagon

There is no standard formula to find out the diagonals of irregular hexagons. In regular hexagons, there are nine diagonals in six equilateral triangles. So, it is easy to determine the length of each diagonal line. If one side of the hexagon is given then it is easy to calculate the all diagonal. It is having diagonals of two different lengths.

Solved Examples for Hexagon Formula

Q.1: Calculate the area and perimeter of a given regular hexagon whose side is 4 cm.

Solution: Given parameters are,

Side of the hexagon, a = 4 cm

Thus area of Hexagon will be,

Solution: Given parameters are:

Perimeter, P = 48 cm

Also, formula for perimeter is,

P=6×a

Hexagon

Fixed number

Definition

A hexagon is a polygon with six sides. A regular hexagon is a regular polygon with six sides and six angles congruent.

Properties

1. Polygon with six sides
2. The regular hexagon has six sides and six angles congruent, with a measure of 120°
3. A regular hexagon can be inscribed into a circle or circumscribed by a circle

Hexagon Formulas

Hexagon Calculator

Welcome to the hexagon calculator, a handy tool when dealing with any regular hexagon. The hexagon shape is one of the most popular shapes in nature, from honeycomb patterns to hexagon tiles for mirrors – its uses are almost endless. Here we explain not only why the 6-sided polygon is so popular but also how to draw hexagon sides correctly. We also answer the question “what is a hexagon?” using the hexagon definition.

With our hexagon calculator, you can explore many geometrical properties and calculations, including how to find the area of a hexagon, as well as teach you how to use the calculator to simplify any analysis involving this 6-sided shape.

How many sides does a hexagon have? Exploring the 6-sided shape

It should be no surprise that the hexagon (also known as the “6-sided polygon”) has precisely six sides. This fact is true for all hexagons since it is their defining feature. The length of the sides can vary even within the same hexagon, except when it comes to the regular hexagon, in which all sides must have equal length.

We will dive a bit deeper into such shape later on when we deal with how to find the area of a hexagon. For now, it suffices to say that the regular hexagon is the most common way to represent a 6-sided polygon and the one most often found in nature.

There will be a whole section dedicated to the important properties of the hexagon shape, but first, we need to know the technical answer to: “What is a hexagon?” Answering this question will help us understand the tricks we can use to calculate the area of a hexagon without using the hexagon area formula blindly. These tricks involve using other polygons such as squares, triangles and even parallelograms.

Hexagon definition, what is a hexagon?

In very much the same way an octagon is defined as having 8 angles, a hexagonal shape is technically defined as having 6 angles which conversely means that (as you can see in the picture above) that the hexagonal shape is always a 6-sided shape. The angles of an arbitrary hexagon can have any value, but they all must sum up to 720º (you can easily convert to other units using our angle conversion calculator).

In a regular hexagon, however, all the hexagon sides and angles must have the same value. For the sides, any value is accepted as long as they are all the same. These restrictions mean that, for a regular hexagon, calculating the perimeter is so easy that you don’t even need to use the perimeter of a polygon calculator if you know a bit of math. Just calculate:

• perimeter = 6 * side

where side refers to the length of any one side.

As for the angles, a regular hexagon requires that all angles are equal and the sum up to 720º, which means that each individual angle must be 120º. This fact proves to be of the utmost importance when we talk about the popularity of the hexagon shape in nature. It will also be helpful when we explain how to find the area of a regular hexagon since we will be using these angles to figure out which triangle calculator we should use, or even which rectangle we will use in another method.

Hexagon area formula: how to find the area of a hexagon

We will now have a look at how to find the area of a hexagon using different tricks. The easiest way is to use our hexagon calculator, which includes a built-in area conversion tool. For those who want to know how to do this by hand, we will explain how to find the area of a regular hexagon with and without the hexagon area formula. The formula for the area of a polygon is always the same no matter how many sides it has as long as it is a regular polygon:

• area = apothem * perimeter / 2

Just as a reminder, the apothem is the distance between the midpoint of any side and the center. You can view it as the height of the equilateral triangle formed by taking one side and two radii of the hexagon (each of the colored areas in the image above). Alternatively, one can also think about the apothem as the distance between the center, and any side of the hexagon since the euclidean distance is defined using a perpendicular line.

If you don’t remember the formula, you can always think about the 6-sided polygon as a collection of 6 triangles. For the regular hexagon, these triangles are equilateral triangles. This fact makes it much easier to calculate their area than if they were isosceles triangles or even 45 45 90 triangles as in the case of an octagon.

For the regular triangle, all sides are of the same length, which is the length of the side of the hexagon they form. We will call this a. And the height of a triangle will be h = √3/2 * a, which is the exact value of the apothem in this case. We remind you that √ means square root. Using this, we can start with the maths:

• A₀ = a * h / 2
• = a * √3/2 * a / 2
• = √3/4 * a²

Where A₀ means the area of each of the equilateral triangles in which we have divided the hexagon. After multiplying this area by six (because we have 6 triangles), we get the hexagon area formula:

• A = 6 * A₀ = 6 * √3/4 * a²
• A = 3 * √3/2 * a²
• = (√3/2 * a) * (6 * a) /2
• = apothem * perimeter /2

We hope you can see how we arrive at the same hexagon area formula we mentioned before.

If you want to get exotic, you can play around with other different shapes. For example, suppose you divide the hexagon in half (from vertex to vertex). In that case, you get two trapezoids, and you can calculate the area of the hexagon as the sum of both, using our trapezoid area calculator. You could also combine two adjacent triangles to construct a total of 3 different rhombuses and calculate the area of each separately. You can even decompose the hexagon in one big rectangle (using the short diagonals) and 2 isosceles triangles!

Feel free to play around with different shapes and calculators to see what other tricks you can come up with. Try to use only right triangles or maybe even special right triangles to calculate the area of a hexagon! Check out the area of the right triangle calculator for help with the computations.

Diagonals of a hexagon

The total number of hexagon’s diagonals is equal to 9 – three of these are long diagonals that cross the central point, and the other six are the so-called “height” of the hexagon.
Our hexagon calculator can also spare you some tedious calculations on the lengths of the hexagon’s diagonals. Here is how you calculate the two types of diagonals:

• Long diagonals – They always cross the central point of the hexagon. As you can notice from the picture above, the length of such a diagonal is equal to two edge lengths:
  D = 2 * a.
• Short diagonals – They do not cross the central point. They are constructed joining two vertices leaving exactly one in between them. Their length is equal to d = √3 * a.

You should be able to check the statement about the long diagonal by visual inspection. However, checking the short diagonals take a bit more ingenuity. But don’t worry, it’s not rocket science, and we are sure that you can calculate it yourself with a bit of time; remember that you can always use the help of calculators such as the right triangle side lengths calculator.

Circumradius and inradius

Another pair of values that are important in a hexagon are the circumradius and the inradius. The circumradius is the radius of the circumference that contains all the vertices of the regular hexagon. The inradius is the radius of the biggest circle contained entirely within the hexagon.

• Circumradius: to find the radius of a circle circumscribed on the regular hexagon, you need to determine the distance between the central point of the hexagon (that is also the center of the circle) and any of the vertices. It is simply equal to R = a.
• Inradius: the radius of a circle inscribed in the regular hexagon is equal to half of its height, which is also the apothem: r = √3/2 * a.

How to draw a hexagon shape

Now we will explore a more practical and less mathematical world: how to draw a hexagon. For a random (irregular) hexagon, the answer is simple: draw any 6-sided shape so that it is a closed polygon, and you’re done. But for a regular hexagon, things are not so easy since we have to make sure all the sides are of the same length.

To get the perfect result, you will need a drawing compass. Draw a circle, and, with the same radius, start making marks along it. Starting at a random point and then making the next mark using the previous one as the anchor point, draw a circle with the compass. You will end up with 6 marks, and if you join them with the straight lines, you will have yourself a regular hexagon. You can see a similar process in the animation above.

The easiest way to find a hexagon side, area…

The hexagon calculator allows you to calculate several interesting parameters of the 6-sided shape that we usually call a hexagon. Using this calculator is as simple as it can possibly get with only one of the parameters needed to calculate all others and includes a built-in length conversion tool for each of them.

We have discussed all the parameters of the calculator, but for the sake of clarity and completeness, we will now go over them briefly:

• Area – 2-D surface enclosed by the hexagon shape;
• Side Length – Distance from one vertex to the next consecutive one;
• Perimeter – Sum of the lengths of all hexagon sides;
• Long Diagonal – Distance from one vertex to the opposite one;
• Short Diagonal – Distance between two vertices that have another vertex between them;
• Circumcircle radius – Distance from the center to a vertex (Same as the radius of the hexagon); and
• Incircle radius – Same as the apothem.

If you like the simplicity of this calculator, we invite you to try our other polygon calculators such as the regular pentagon calculator or even 3-D calculators such as the pyramid calculator, triangular prism calculator, or the rectangular prism calculator.

Hexagon tiles and real-world uses of the 6-sided polygon

Everyone loves a good real-world application, and hexagons are definitely one of the most used polygons in the world. Starting with human usages, the easiest (and probably least exciting) use is hexagon tiles for flooring purposes. The hexagon is an excellent shape because it perfectly fits with one another to cover any desired area. If you’re interested in such a use, we recommend the flooring calculator and the square footage calculator as they are excellent tools for this purpose.

The next case is common to all polygons, but it is still interesting to see. In photography, the opening of the sensor almost always has a polygonal shape. This part of the camera is called the aperture and dictates many properties and features of the pictures produced by a camera. The most unexpected one is the shape of very bright (point-like) objects due to the effect called diffraction grating, and it is illustrated in the picture above.

One of the most valuable uses of hexagons in the modern era, closely related to the one we’ve talked about in photography, is in astronomy. One of the biggest problems we experience when observing distant stars is how faint they are in the night sky. That is because despite being very bright objects, they are so very far away that only a tiny fraction of their light reaches us; you can learn more about that in our luminosity calculator. On top of that, due to relativistic effects (similar to time dilation and length contraction), their light arrives on the Earth with less energy than it was emitted. This effect is called the red shift.

The result is that we get a tiny amount of energy with a longer wavelength than we would like. The best way to counteract this is to build telescopes as enormous as possible. The problem is that making a one-piece lens or mirror larger than a couple of meters is almost impossible, not to talk about the issues with logistics. The solution is to build a modular mirror using hexagonal tiles like the ones you can see in the pictures above.

Making such a big mirror improves the angular resolution of the telescope, as well as the magnification factor due to the geometrical properties of a “Cassegrain telescope”. So we can say that thanks to regular hexagons, we can see better, further, and more clearly than we could have ever done with only one-piece lenses or mirrors.

Did you know that hexagon quilts are also a thing??

Hexagon Formula

A polygon is a two-dimensional (2-D) closed figure made up of straight line segments. In geometry, hexagon is a polygon with 6 sides. If the lengths of all the sides and the measurement of all the angles are equal, such hexagon is called a regular hexagon. In other words, sides of a regular hexagon are congruent.

There is a predefined set of formulas for the calculation of perimeter and area of a regular hexagon which is collectively called as hexagon formula. The hexagon formula for a hexagon with the side length of a, is given as:

Hexagon formula helps us to compute the area and perimeter of hexagonal objects. Honeycomb, quartz crystal, bolt head, Lug/wheel nut, Allen wrench, floor tiles etc are few things which you would find a hexagon. 

Properties of a Regular Hexagon:

• It has six sides and six angles.
• Lengths of all the sides and the measurement of all the angles are equal.
• The total number of diagonals in a regular hexagon is 9.
• The sum of all interior angles is equal to 720 degrees, where each interior angle measures 120 degrees.
• The sum of all exterior angles is equal to 360 degrees, where each exterior angle measures 60 degrees.

Derivation:

Consider a regular hexagon with each side a units.

Formula for area of a hexagon: Area of a hexagon is defined as the region occupied inside the boundary of a hexagon.

In order to calculate the area of a hexagon, we divide it into small six isosceles triangles. Calculate the area of one of the triangles and then we can multiply by 6 to find the total area of the polygon.

Take one of the triangles and draw a line from the apex to the midpoint of the base to form a right angle. The base of the triangle is a, the side length of the polygon. Let the length of this line be h.

The sum of all exterior angles is equal to 360 degrees. Here, ∠AOB = 360/6 = 60°

∴ θ = 30°

We know that the tan of an angle is opposite side by adjacent side,

Formula for perimeter of a hexagon: Perimeter of a hexagon is defined as the length of the boundary of the hexagon. So perimeter will be the sum of the length of all sides. The formula for perimeter of a hexagon is given by:

Perimeter = length of 6 sides

Perimeter of an Hexagon = 6a

Solved examples:

Question 1: Calculate the area and perimeter of a regular hexagon whose side is 4.1cm.

Solution: Given, side of the hexagon = 4.1 cm

Area Of Hexagon Formula

A polygon with six sides and six angles is termed as a hexagon. Similarly, we have Pentagon where the polygon has 5 sides; Octagon has 8 sides. Each internal angle of the hexagon has been calculated to be 120°.

In general, the sum of interior angles of a Polygon is given by-

(n−2)×180°

Types of Hexagon

A hexagon can be of two types, namely

• Regular Hexagon
• Irregular Hexagon

In the case of a regular hexagon, all the sides are of equal length, and the internal angles are of the same value. The regular hexagon consists of six symmetrical lines and rotational symmetry of order of 6.

Whereas in the case of the irregular hexagon, neither the sides are equal, nor the angles are the same.

Formula for the Area of a Hexagon

Area of the hexagon is the space confined within the sides of the polygon.

The area of Hexagon is given by

Area of a Hexagon Problems

Question 1: 
Find the area of a regular hexagon whose side is 4 cm?

Solution:
Given
s = 4 cm

By putting the value of s, we get:

A = 41.57 cm. sq.

✅ Math Formulas ⭐️⭐️⭐️⭐️⭐