Regular Hexagon

What is Regular Hexagon?

Before jumping into Hexagon, we must know what Polygons are. Polygons are a part of our everyday life. We see and use a lot of things which are of different shapes. These shapes are all different forms of a polygon. The meaning of the term ‘poly’ is ‘many’ and ‘gon’ means ‘angle’. Therefore, polygons are two-dimensional closed figures which are formed using more than three straight lines and angles. The classification of polygons is done based on the number of sides and vertices it has. For example, a polygon with 3 sides and 3 angles is known as Triangle whereas a polygon with 4 sides and 4 angles is known as a quadrilateral. A polygon is formed with only straight lines. Any closed figure with curved lines is not considered as a polygon. The area and the perimeter of a polygon depend upon its types. Now that we know what is a polygon, I am sure that now you have some idea about what is a regular hexagon?

A Hexagon is a polygon having 6 sides and 6 vertices. The sum of the internal angles of a regular hexagon is always 720° and can be calculated by counting the number of triangles fitting inside the hexagon. 

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Since there are 4 triangles and the sum of each internal angle of a triangle is 180°, the 4 triangles, side by side, would measure up to 4x180=720°.

Types of Hexagon

Hexagons can be classified as:

1. Regular or irregular: A hexagon with all its sides and angles equal is known as regular hexagon whereas a hexagon which is not regular is called an irregular hexagon.  

2. Convex or concave: If a hexagon is convex, none of its interior angles would be greater than 180° whereas a concave hexagon has one or more of its interior angles greater than 180°. 

Regular Hexagon

A regular hexagon is equilateral as well as equiangular. It is bicentric, which means that it is both cyclic (that has a circumscribed circle) and tangential (that has an inscribed circle). A regular hexagon has 6 rotational symmetries as well as 6 reflection symmetries. 

How to Draw a Regular Hexagon

A regular hexagon of a given side length can be drawn by just using only a ruler and a compass and by following few steps given below:

1. First, we need to draw a linear segment whose length will be the desired hexagon side length.

2. Next, we need to construct a circle, having its centre at one end of the linear segment and radius equal to the segment length.

3. Construct a second circle without changing the radius, while constructing the second circle consider its centre at the other end of the linear segment. The two points are defined where this circle intersects with the first one.

4. Using the same radius, place the tip of the compass on the last defined point and construct a new circle. A new point will be defined as it intersects with the first circle.

5. This same procedure will be repeated two more times. Place the tip of the compass on the last point and construct another circle. 

6. There will be a total of six points that will be defined so far around the first circle. These will become the vertices of the hexagon. Next thing that needs to be done is to draw linear segments between them and there…. your regular hexagon is ready. The initial circle will be the circumcircle of the hexagon.

Given Below is the Stepwise Drawing Procedure of the Hexagon.

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Circumcircle and Incircle of a Regular Hexagon

In a regular polygon, a circle can be drawn that passes through all six vertices of the hexagon. This circle is known as the circumscribed circle or can be called a circumcircle of the polygon. The centre of the hexagon will be the centre of this circle. In the same way, the diagonals of the hexagon will be the diameters of the circumcircle. The radius of circumcircle Rcis called circumradius.

The circle that can be drawn inside the hexagon that passes through the middles of its sides can be called an inscribed circle or an incircle. All the six sides of the hexagon form the tangent of this circle. The centre of the hexagon coincides with the centre of the incircle. The radius of incircle Ri is called inradius.

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Perimeter and Area of Regular Hexagon

The perimeter of a regular hexagon can be determined by calculating the sum of all the six sides of a hexagon. we can also use a (regular hexagon formula) formula to calculate its perimeter.

Perimeter of a hexagon (P) = 6 × sides(a)

                                 = 6a

The area of a regular hexagon can be determined using three different ways with a formula for each:

1. When the length of the side of a regular hexagon is given, the formula will be:

                        Area = s²n/4 tan(180/n)

     Where, s = length of the side 

            n = number of the side

2. When the radius (circumradius) is given, the formula will be:

                        Area = r²n sin(360/n)/2

  Where, r = radius (circumradius) and n =  number of the side

3. When the apothem (inradius) is given, the formula will be:

                      Area = a²n tan(180/n)

     Where, a = length of the apothem                 

                    n = number of the side

      Properties of a Regular Hexagon

1. All the sides are equal in length.

2. The radius of a hexagon is equal to the length of its sides. 

3. The interior angles of a regular hexagon are 120⁰ 

4. The exterior angles of a regular hexagon are 60⁰

5. Sum of interior angle is 720⁰ 

Solved Examples

Q1. What will be the Area of a Regular Hexagon with an Inradius of 10√3?

Solution 1. First we have to find the perimeter of the hexagon. Use the apothem to find the perimeter of the hexagon. The perimeter will be 120. Now using the formula to find the area of the hexagon

Area = 1/2pa

         = ½ x 120 x 10√3

         = 600√3 units²