Hexagon Formula: Definition, Derivation and Examples

Hexagons are polygons that have six sides. Regular hexagons, irregular hexagons, and concave hexagons are the three types of hexagons. The hexagon formula is a series of formulas for calculating the hexagon’s perimeter, area, and diagonals.

In this article, we will learn about the definition of the hexagon, properties of a hexagon, different types of hexagons and formulas to calculate the area and perimeter of a regular pentagon.

What is a Hexagon?

A hexagon is a six-sided polygon with six straight line segments and six interior angles, which add up to \({720^ \circ }.\) A simple hexagon required six straight sides that meet to produce six vertices.

The word hexagon comes from the Greek word Hexa, which means six, and gonia, which means angle.
Some real-life examples of the hexagon are a honeycomb,  a hexagonal floor tile, a hexagonal wall clock, etc.

Types of Hexagon

We can classify a polygon as a concave polygon, convex polygon, regular polygon, and irregular polygon based on the sides and angles. Similarly, the hexagon has four types. They are,

1. Concave Hexagon
2. Convex Hexagon
3. Regular Hexagon
4. Irregular Hexagon

Concave Hexagon

A hexagon in which at least one angle is more than \({180^ \circ }\) is called a concave hexagon. In other words, if the vertices are pointed towards inwards of a hexagon, then it is known as a concave hexagon.

Convex Hexagon

A hexagon in which all angles are less than \({180^ \circ }\) is called a convex hexagon. In other words, if the vertices point outwards of a hexagon, then it is known as a convex hexagon.

Regular Hexagon

A regular hexagon is one that has all of its sides equal and all of its angles congruent.

Irregular Hexagon

Hexagons that aren’t regular or a hexagon in which the sides and angles are unequal is known as an irregular hexagon.

Angles in a Regular Hexagon

We have general formulas to find the interior and exterior angles of a polygon. If we consider a regular polygon of \(n\) sides, we have
The sum of interior angles \( = \left({n – 2} \right) \times {180^ \circ }\)
The measure of each interior angle \( = \frac{{\left({n – 2} \right) \times {{180}^ \circ }}}{n}\)
For any polygon, the sum of the measure of exterior angles \( = {360^ \circ }\)
And so, the measure of each exterior angle \( = \frac{{{{360}^ \circ }}}{n}\)
We can apply these formulas to find the interior and exterior angles of the regular hexagon.
The sum of interior angles of a hexagon \( = \left({6 – 2} \right) \times {180^ \circ }\)
\( = 4 \times {180^ \circ }\)
\( = {720^ \circ }\)
The measure of each interior angle of a regular hexagon \( = \frac{{{{720}^ \circ }}}{6} = {120^ \circ }\)
The sum of the exterior angles of a hexagon \( = {360^ \circ }\)
The measure of each interior angle of a regular hexagon \( = \frac{{{{360}^ \circ }}}{6} = {60^ \circ }\)

Diagonals of a Hexagon

A diagonal is a line segment that connects the two non-adjacent sides of a polygon.
The formula computes the number of diagonals of a polygon is given by
Number of diagonals \( = \frac{{n\left({n – 3} \right)}}{2}\)
Where \(n\) is the number of sides of a polygon.
So, the number of diagonals in a hexagon \( = \frac{{6\left({6 – 3} \right)}}{2} = \frac{{18}}{2} = 9\)

In the hexagon \(ABCDEF,\) the non adjacent vertices are \(A\) and \(D,A\) and \(C,A\) and \(E,B\) and \(F,B\) and \(E,B\) and \(D,C\) and \(F,C\) and \(E,D\) and \(F.\)

If we join \(A\) and \(D,\) then we will get the diagonal \(AD.\) Joining \(A\) and \(C\) we get the diagonal \(AC,\) joining \(A\) and \(E,\) we get the diagonal \(AE,\) joining \(B\) and \(F,\) we get the diagonal \(BF,\) joining \(B\) and \(E,\) we get the diagonal \(BE,\) joining \(B\) and \(D,\) we get the diagonal \(BD,\) joining \(C\) and \(F,\) we get the diagonal \(CF.\) Similarly, joining \(C\) and \(E,\) we will get the diagonal \(CE\) and joining \(D\) and \(F,\) we get the diagonal \(DF.\)

Properties of Hexagon

The properties of a hexagon are given below.

1. The number of diagonals in a hexagon is nine.
2. A regular hexagon is a convex hexagon as the sum of all its internal angles are less than \({180^ \circ }.\)
3. A regular hexagon can be divided into six equilateral triangles
4. A regular hexagon has equal sides so, and it is symmetrical.
5. The opposite sides of a regular hexagon are always parallel to each other.

Perimeter of a Hexagon

The sum of the length of the boundary of any closed figure is its perimeter. The perimeter of a regular or irregular hexagon will be the sum of the lengths of its sides.

If \(a,b,c,d,e,f\) are respectively the lengths of sides of an irregular hexagon, then the perimeter of an irregular hexagon is
Perimeter \( = a + b + c + d + e + f\)
If \(a\) represents the side of a regular hexagon, then the
The perimeter of a regular hexagon \( = 6a\)

Area of a Hexagon

The area of a regular hexagon is the region or the space occupied by the shape. We measure the area in square units. We can calculate the area of the hexagon by dividing it into equilateral triangles. 

We can divide it into six triangles. If we know the area of one equilateral triangle, we can easily calculate the area of the hexagon.

We know that the area of an equilateral triangle is \(\frac{{\sqrt 3 {a^2}}}{4}\,{\rm{square}}\,{\rm{units}},\) where \(a\) is the side length of the equilateral triangle.
Hence, we can calculate the area of the hexagon if we combine six such triangles, which is \(6 \times \frac{{\sqrt 3 {a^2}}}{4} = \frac{{3\sqrt 3 {a^2}}}{2}{\text{square}}\,{\text{units}}.\)

Solved Examples

Q.1. Find the area of the given regular hexagon whose side measure is \(6\,{\text{cm}}.\)
Ans: We know that the area of a regular hexagon with side measure \(a\,{\text{units}}\) is given by
\(A = \frac{{3\sqrt 3 {a^2}}}{2}{\text{square}}\,{\text{units}}\)
So, the area of a regular hexagon with a side measure of \(6\,{\text{cm}}\) is \(A = \frac{{3\sqrt 3 \times {{\left( 6 \right)}^2}}}{2}{\text{sq}}{\text{.cm}}\)
\(\frac{{3\sqrt 3 \times 36}}{2} = 3\sqrt 3 \times 18\)
\( = 93.53\,{\text{c}}{{\text{m}}^2}\) (approx)
Therefore, the area of the regular polygon with a side measure of \(6\,{\text{cm}}\) is \( = 93.53\,{\text{c}}{{\text{m}}^2}\) (approx)

Q.2. Find the perimeter of the given regular hexagon whose side measure is \(5\,{\text{cm}}.\)
Ans: If a represents the side of a regular hexagon, then the perimeter of a regular hexagon \( = 6a\)
Therefore, the perimeter of a regular hexagon with a side of \(5\,{\text{cm}}\) is \(5 \times 6 = 30\,{\text{cm}}.\)

Q.3. Determine the length of the sides of a regular hexagon, if the hexagon’s area is \({\text{300}}\sqrt 3 \,{\text{square}}\,{\text{units}}.\)
Ans: We know that the area of a regular hexagon with side measure \(a\,{\text{units}}\) is given by
\(A = \frac{{3\sqrt 3 {a^2}}}{2}{\text{square}}\,{\text{units}}\)
Therefore, \(300\sqrt 3 = \frac{{3\sqrt 3 {a^2}}}{2}\)
Cancelling \(\sqrt 3 \) on both sides we have,
\(\frac{3}{2}{a^2} = 300\)
\( \Rightarrow {a^2} = 300 \times \frac{2}{3}\)
\( \Rightarrow a = \sqrt {200} = 10\sqrt 2 \,{\text{units}}.\)
Therefore, the length of the sides is \(10\sqrt 2 \,{\text{units}}.\)

Q.4. Evaluate the length of the side of a regular hexagon if its perimeter is given as \(60\,{\text{cm}}.\)
Ans: Given that the perimeter \( = 60\,{\text{cm}}\)
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,
\(a = \frac{{60}}{6} = 10\,{\text{cm}}\)
Hence, the length of the side of the hexagon is \(10\,{\text{cm}}.\)

Q.5. Identify hexagon from the following figures.
Figure 1:

Figure 2:

Ans: In figure \(1,\) we can see a polygon with six sides and in figure \(2,\) a polygon with five sides.
A pentagon is a five-sided polygon with five straight lines and five interior angles. So, figure \(2\) is a pentagon.
A hexagon is a six-sided polygon with six straight lines and six interior angles.
Therefore, figure \(1\) is a hexagon.

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.

A few important hexagon notes are:

1. It has six edges, six vertices, and six sides.
2. In terms of measurement, all of the side lengths are equal or unequal.
3. In a regular hexagon, all internal angles are equal to \({120^ \circ }.\)
4. The sum of internal angles is always to \({720^ \circ }.\)

Learn All the Concepts on Diagonal Formula

FAQs

Q.1. What is the formula for the area of a hexagon?
Ans: The formula of the area of a regular hexagon with side measure \(a\,{\text{units}}\) is given by \(A = \frac{{3\sqrt 3 {a^2}}}{2}{\text{square}}\,{\text{units}}\)

Q.2. How to find the area of a hexagon formula?
Ans: We can calculate the area of a hexagon by dividing it into equilateral triangles. We can divide it into six triangles. We know that the area of an equilateral triangle is \(\frac{{\sqrt 3 {a^2}}}{2}{\text{square}}\,{\text{units,}}\) where \(a\) is the side length of the equilateral triangle.
Hence, the area of the hexagon we can calculate if we combine six such triangles, which is \(6 \times \frac{{\sqrt 3 {a^2}}}{4} = \frac{{3\sqrt 3 {a^2}}}{2}{\text{square}}\,{\text{units}}{\text{.}}\)

Q.3. What are the types of the hexagon?
Ans: The types of the hexagon are,
1. Concave hexagon
2. Convex hexagon
3. Regular hexagon
4. Irregular hexagon

Q.4. What do you mean by a regular hexagon and an irregular hexagon?
Ans: A hexagon with all sides equal and equal angles is called a regular hexagon.
Hexagons that aren’t regular are called irregular hexagons. A hexagon in which the sides and angles are unequal is known as an irregular hexagon.

Q.5. What is the interior and exterior angles of a regular hexagon?
Ans: The sum of interior angles of a hexagon \( = \left({6 – 2} \right) \times {180^ \circ }\)
\( = 4 \times {180^ \circ }\)
\( = {720^ \circ }\)
The measure of each interior angle of a regular hexagon \( = \frac{{{{720}^ \circ }}}{6} = {120^ \circ }\)
The sum of the exterior angles of a hexagon \( = {360^ \circ }\)
The measure of each interior angle of a regular hexagon \( = \frac{{{{360}^ \circ }}}{6} = {60^ \circ }\)