Perimeter of Hexagon Formula: Definition, Derivation and Examples

A hexagon is a polygon with six sides and six angles. The perimeter of a polygon is the sum of the measures of its sides. Therefore, the perimeter of a hexagon is the sum of the lengths of six sides of it. A hexagon with six equal sides is called a regular hexagon. We might need to find the perimeter when putting up lights or fencing the hexagonal shaped garden, constructing a hexagonal-shaped building, building a hexagonal pool, etc. In this article, we will learn the definition of hexagon, Perimeter of Hexagon Formula, properties of a hexagon, different types of hexagons and formulas to calculate the perimeter of hexagon. Read on to learn more.

Perimeter of Hexagon Formula

Hexa comes from the Greek word “Hex” which means “six” in English and “gonia” meaning angles. Therefore, a hexagon has six sides, angles, and vertices. A hexagon with six equal sides is called a regular hexagon. The formula for the Perimeter of Hexagon: P=6×a
Where,
a = Length of one side

Types of Hexagon

There are four types of polygons based on their angles and sides: regular polygon, irregular polygon, concave polygon, convex polygon. Similarly, a hexagon is divided into four categories. They are regular hexagon, irregular hexagon, concave hexagon and convex hexagon.

Regular Hexagon

A hexagon that has six equal sides and six equal angles is called a regular hexagon.

Irregular Hexagon

A hexagon that has unequal sides and angles is called an irregular hexagon.

Concave Hexagon

A hexagon that has at least one angle of more than \({180^ \circ }\) is called a concave hexagon. If the vertices point within or pointing inside in a hexagon, it is known as a concave hexagon.

Learn All the Important Hexagon Formulas

Convex Hexagon

A hexagon in which each angle is less than \({180^ \circ }\) is called a convex hexagon. In other words, if the vertices point outwards in a hexagon or pointing outside is known as a convex hexagon.

Interior and Exterior Angles of a Regular Hexagon

We know that, for a regular polygon of \(n\) sides, we have 

(i) Each exterior angle
(ii) Each interior angle \( = 180^\circ  – \) (each exterior angle)

\( = {180^ \circ } – \frac{{{{360}^ \circ }}}{n}\)
\( = \frac{{n \times {{180}^ \circ } – \left( {2 \times {{180}^ \circ }} \right)}}{n}\)
\( = \frac{{(n – 2) \times {{180}^ \circ }}}{n}\)

So, the sum of the exterior angles of a regular hexagon \( = {360^ \circ }\)

The measure of each exterior angle of a regular hexagon \( = \frac{{{{360}^ \circ }}}{6} = {60^ \circ }\)

The sum of the interior angles of a regular hexagon \( = {720^ \circ }\)

And the interior angle of a regular hexagon \( = \frac{{4 \times {{180}^ \circ }}}{6} = {120^ \circ }\)

Properties of Hexagon

1. The sum of all interior angles of a regular hexagon is \({720^ \circ }\), and the sum of exterior angles is \({360^ \circ }\)
2. The measure of each interior angle in regular hexagon measures \({120^ \circ }\), and the measure of each exterior angle in a hexagon measures \({60^ \circ }\)
3. The total number of diagonals in a hexagon is \(9.\)
4. We can divide a regular hexagon into six equilateral triangles.
5. The opposite sides of a regular hexagon are always parallel to each other.

Perimeter

The word perimeter has been introduced from the Greek word peri, which means around, and metron, which means measure.

The total length of the edges of a figure is known as the perimeter. For any polygon, the perimeter is the sum of the side length of it. For any closed figure except polygon, perimeter means the length of its boundary or the outer line.

Perimeter of a Hexagon

The perimeter of a regular or irregular hexagon means the sum of the lengths of its edges. As a hexagon has six sides, the hexagon’s perimeter will be the sum of the measure of all these six sides.

If \(a, b, c, d, e,f\) are respectively the edges of an irregular hexagon, then the perimeter of an irregular hexagon is Perimeter \(=a+b+c+d+e+f\)

For a regular hexagon, as the measure of all sides is equal, the perimeter is the product of the number of sides and measurement of each side. If the measure of a side of a regular hexagon is a then 

The perimeter of a regular hexagon \(=6a\)

Examples of Hexagons

We use hexagonal shapes in our daily life, such as hexagonal honeycomb, hexagonal pencils etc. Even we can see hexagons on the outer cover of footballs. A hexagon is either regular or irregular.

A simple hexagon requires six straight sides that meet to produce six vertices.

What is a Polygon?

The word polygon is taken from the Greek word poly, which means many, and gon means angle. A polygon is a simple closed curve formed by three or more line segments such that

(i) two-line segments do not intersect except at their endpoints.
(ii) two line segments with a common endpoint are not coincident.

In other words, a polygon is a simple closed two-dimensional shape formed by joining the straight line segments. For example, squares, triangles, rectangles, pentagons, hexagons etc., are polygons.

Is a circle a polygon? No, it is not a polygon as line segments do not make it, but it is known as a simple closed curve.

Each straight line segment in a polygon is called its side. A triangle, quadrilateral, pentagon, hexagon, heptagon, octagon, nonagon, and decagon are called a polygon accordingly as it contains \(3,4, 5, 6, 7, 8, 9, 10\) sides, respectively.

Solved Examples – Hexagon Formula

Q.1. Find the perimeter of the regular hexagon whose area is \(150\sqrt 3 \;{\rm{c}}{{\rm{m}}^2}\)
Ans: We know that the area of a regular hexagon with side measure a units is given by \(A = \frac{{3\sqrt 3 {a^2}}}{2}\) square units So, the area of a regular hexagon is \(\frac{{3\sqrt 3 {{(a)}^2}}}{2} = 150\sqrt 3 \)
Now, cancelling \(\sqrt 3 \) from both sides we have,
\(\frac{{3{{(a)}^2}}}{2} = 150\)
\( \Rightarrow {(a)^2} = \frac{{150 \times 2}}{3} = 100\)
\( \Rightarrow a = \sqrt {100} = 10\;{\rm{cm}}\)
Therefore, the perimeter of the hexagon is \(10 \times 6 = 60\;{\rm{cm}}\)

Q.2. Find the perimeter of the given regular hexagon whose side measure is 15 cm
Ans: If a represents the side of a regular hexagon, then the perimeter of a regular hexagon \({\rm{ = 6a}}\)
Therefore, the perimeter of a regular hexagon with a side of \({\rm{15\,cm}}\) is \({\rm{15 \times 6 = 90\;cm}}\)

Q.3. Determine the length of the sides of a regular hexagon, if its perimeter is 18 cm. 
Ans: Given that the perimeter \({\rm{18\,cm}}\)
If a represents the side of a regular hexagon, then the perimeter of a regular hexagon \({\rm{ = 6a}}\)
According to the question,
\(6a=18\)
Now, transposing \(6\) into RHS we have,
\(a = \frac{{18}}{6} = 3\;{\rm{cm}}\) 
Hence, the length of the side of the hexagon is \(3\;{\rm{cm}}\)

Q.4. Evaluate the length of the side of a regular hexagon if its perimeter is given as 60 cm
Ans: Given that the perimeter \({\rm{ = 60\,cm}}\)
If a represents the side of a regular hexagon, then the perimeter of a regular hexagon \(=6a\)
According to the question,
\(6a=60\)
Now, transposing \(6\) into RHS we have,
\(a = \frac{{60}}{6} = 10\;{\rm{cm}}\)
Hence, the length of the side of the hexagon is \(10\;{\rm{cm}}\)

Q.5. Find the missing length if the perimeter of the concave hexagon is 20.5 cm

Ans: We can observe that the above hexagon is a concave hexagon. The perimeter of the given hexagon is \(20.5\) units.
The lengths of the polygon sides are \(AB=3\) units, \(BC=4\) units, \(CD=6\) units, \(DE=2\) units, \(EF=1.5\) units and \(FA = x\) units.
It is known that the perimeter of the polygon \(ABCDEF = 20.5\) units
\( \Rightarrow \) Perimeter of polygon \(ABCDEF=AB+BC+CD+DE+EF+FA=20.5 \) units \(\Rightarrow(3+4+6+2+1.5+x)\) units \(=20.5\) units
Thus, \(x=20.5-3+4+6+2+1.5=4\) units
Therefore, the missing length of the irregular hexagon is \(4\) units.

Summary

In this article, we have learned the definition of hexagon, different types of hexagon, the formula to find the hexagon area when one side of the hexagon is given, and the formula to find the perimeter of a hexagon. Also, we have solved some example problems based on the formula of area and perimeter of the hexagon.

Learn All the Important Polygon Formulas

Frequently Asked Questions (FAQs) – Hexagon Formula

The answers to the most commonly raised doubts on perimeter of hexagon formula are given below:

We hope this detailed article on the perimeter of hexagon formula helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!