Hexagon Area Formula

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.

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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

FAQ (Frequently Asked Questions)

Q1. How to Calculate from an Irregular Hexagon?

One way to calculate the area of an Irregular Hexagon is with given vertices. That said, if you only know the vertices of the hexagon, you need to first create a chart with 2 columns and 7 rows. Next, label each row of the six points like (Point A, Point B, Point C, Point D, etc), and each column as the ‘x’ or ‘y’ coordinates of those points. Index the x and y coordinates of Point A to the right of Point A, Point B to the right of Point B, and so forth. Repeat the coordinates of the first point at the bottom of the index. Suppose, you’re working with the following points, in (x, y) pattern:-

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Multiply in x coordinates

A: (2, 6) [2 x 5] = 10

B: (1, 5) [1 x 8] = 8

C: (3, 8) [3 x 4] = 12

D: (7, 4) [7 x 6] = 42

E: (6, 6) [6 x 1] = 6

F: (10, 1) [10 x 6] = 60

A (again): (2, 6)

Summation is 138

Now, multiply in y coordinates

A: (2, 6) [6 x 1] = 6

B: (1, 5) [5 x 3] = 15

C: (3, 8) [8 x 7] = 56

D: (7, 4) [4 x 6] = 24

E: (6, 6) [6 x 10] = 60

F: (10, 1) [1 x 2] = 2

Summation is 163

Next, subtract the sum of the 2nd group from the sum of the 1st group of coordinates. Subtracting 163 from 138 = -25. Take the absolute value.

Now, dividing 25 by 2 will give you the Area of an irregular hexagon

25 / 2 = 12.5 square units ( area of an irregular hexagon is in square units)

Q2. Are Hexagon Tiles Used in Daily Life?

Hexagons are actually one of the most used polygons in the real world applications. These are used in tiles for flooring purposes. The hexagon is considered to be an excellent shape as it perfectly fits to cover any desired area.

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