What are Regular Polygons?
Polygons that are equiangular and equilateral are regular polygons. Here is a list of regular polygons:
What is a Regular Hexagon?
A closed-shape polygon with six equal sides and six equal angles is called a regular hexagon.
Properties of a Regular Hexagon
- It has 6 sides of the same size and 6 angles of equal measure.
- It has a total of six vertices.
- The measure of each interior angle is 120° and the measure of each exterior angle is 60°.
- The angle sum of all the interior angles of a hexagon is 720°. The formula for finding the angle sum of a polygon is:
= (n – 2) x 180°, where n is the number of sides of a polygon.
- A regular hexagon can be divided into 6 equilateral triangles.
- The number of diagonals in a hexagon is 9. This can be found using the formula:
\(n~\times~\frac{(n-3)}{2}\), where n is the number of sides of a polygon
- The opposite sides of a regular hexagon are parallel to each other.
The Area and Perimeter of a Regular Hexagon
Now that we have discussed the definition and properties of a regular hexagon, let us look at its area and perimeter.
The area of a regular hexagon is calculated as follows:
\(A=\frac{3\sqrt 3}{2}~\times~a^2\)
or,
\(A=2.59807a^2\)
Where a is the length of one of its sides.
The perimeter of the hexagon is:
P = 6a
Where a denotes the length of a side.
The Angles of a Regular Hexagon
The sum of the angles of a regular hexagon with all six sides equal is 720°, as we already know.
As a result, the interior and exterior angles are as follows:
Each interior angle is equal to
\(\frac{720^{\circ}}{6}=120^{\circ}\)
Each exterior angle = 180° – interior angle
= 180° – 120°
= 60°
Solved Regular Hexagon Examples
Example 1:
What is the area of a regular hexagon that has a side of 3 units?
Solution:
As we know, the area of a regular hexagon \(A=\frac{3\sqrt 3}{2}~\times~a^2\) square units.
Given, the side of a regular hexagon ‘a‘ = 3 units.
Therefore, the area is:
\(A=\frac{3\sqrt 3}{2}~\times~3^2\)
\(=\frac{3\sqrt 3}{2}~\times~9\)
\(=\frac{27\sqrt 3}{2}\) Substitue \(\sqrt 3=1.732\) and simplify.
= 23.38 square units
As a result, the required area is 23.38 square units.
Example 2:
Calculate the perimeter of a regular hexagon with 30 cm long sides.
Solution:
It is given that the side of a regular hexagon ‘a‘ =30 cm.
The perimeter of a regular hexagon = 6 x a
= 6 × 30 cm
= 180 cm
As a result, the perimeter of the hexagon is 180 cm.
Example 3:
If the perimeter of a regular hexagon is 108 units, what is the length of each side?
Solution:
The perimeter of the regular hexagon = 108 units.
The length of the sides can be calculated as
\(\frac{Perimeter}{6}\)
= \(\frac{108}{6}\)
= 18 units
As a result, each side of the hexagon measures 18 units in length.
Hexagons have six sides, six angles, and six vertices.
There are 9 diagonals in a hexagon.
The sum of the interior angles of a hexagon is \(720^{\circ}\).
A hexagon, if not regular, does not have all sides of equal length. All sides of a regular hexagon are of the same length.
Hexagons can be classified as:
- Regular Hexagon
- Irregular Hexagon
- Concave Hexagon
- Convex Hexagon
The measure of all the interior angles of a concave polygon is less than 180 degrees. In contrast, a concave polygon has at least one interior angle having a measure of more than 180 degrees.