Polygon? Regular Polygon? Hexagon? Regular Hexagon? A lot of mathematical terms! You do not know what exactly they are and you are probably confused. Do not worry because we are going to make an end to that confusion today. These words might seem scary but in actual they are not. You might be familiar with a lot of the concepts, you just do not know what the terms mean. First, we will see what is a polygon, then we will move on to what regular and irregular polygons are and then we will get into the details of a regular hexagon. In this post you will also learn to calculate the area, perimeter and sum of internal angles of polygon along with few solved examples.
What is a polygon
You have been studying geometrical shapes like squares, rectangles, and triangles, etc since childhood, right? These are examples of polygons. So, polygons are 2-dimensional(plane), closed figures with straight sides. Closed figures are those figures in which exactly two sides meet at the vertex or corner.
- A polygon has always more than 2 sides.
- Is the circle a polygon?
No, because it does not consist of straight sides.
- Is the box a polygon?
No, because it is a 3-dimensional figure.
Since you know what polygons are, let us study types of polygons.
Different types of Polygons
There are two types of polygons:
- Regular Polygon
- Irregular Polygon
What is an irregular polygon
An irregular polygon is a polygon that can have sides of unequal length and angles of any measure. For example, rectangle, scalene triangle, etc. They have unequal lengths and angles.
What is a regular polygon
A regular polygon is a polygon that has all the sides of equal lengths and equal angles.
Examples of regular polygons include equilateral triangle, square, pentagon, hexagon, etc. All of them have are equiangular (equal angles) and equilateral (same sides).
Now you know the definition of regular and irregular polygon. We are going to continue our study of regular polygons taking hexagon as an example to explain its properties.
A Hexagon is a polygon that has six sides and thus six vertices or corners. Irregular hexagon has lengths and angles of any measure. However, regular hexagon has the same lengths and angles.
We are going to study regular hexagons here. From now onwards, we mean regular hexagon whenever we refer to just hexagon.
If the perimeter of hexagon (i.e. total length of the boundary of hexagon) is P then the length of each side is P/6.
Let us now move on to angles.
Angles of a polygon
There are two types of angles in polygons.
- Exterior angle
- Interior angle
Exterior angles of a polygon
An exterior angle is an angle that is obtained by the side of the polygon and by the extension of the adjacent side.
The sum of the exterior angles in a polygon is always 360o or 2π radians. In case of a regular polygon, the value of each exterior angle can be found by dividing 360o by the number of sides.
- For example, there are 6 sides in a hexagon and all the exterior angles are equal. So, each exterior angle can be found as follows:
Exterior angle = 360o / 6 = 60o or π/3 radians
- Similarly, the value of each exterior angle of a regular heptagon ( a polygon with 7 sides ) can be found by 360o divided by 7.
Interior angles of a polygon
An interior angle is an angle that is inside the polygon i.e. between any two sides, at the vertex.
The interior angles and exterior angles are measured along the same axis. So, if the exterior angle is 60o, then,
Interior angle = 180o – 60o = 120o or 2π/3 radians.
All the interior angles are equal for the regular hexagon. Therefore,
The sum of all the interior angles = 6 × 120o = 720o or 4π radians.
But there is a shortcut to find the sum of the interior angle of a polygon and that is by using a direct formula.
Polygon interior angle formula
Formula of the sum of interior angle of a polygon = ( N – 2 ) * 180o
Where ‘N’ is the number of sides.
Let we calculate the sum of the interior angle of few polygons.
- Hexagon , a polygon with 6 sides : ( 6 – 2 ) * 180o = 720o
- Heptagon, a polygon with 7 sides: ( 7-2 ) * 180o = 900o
- Octagon, a polygon with 8 sides: ( 8-2 ) * 180o = 1080o
- Nonagon, a polygon with 9 sides: ( 9 – 2 ) * 180o = 1260o
Area of a polygon
There is no common formula applicable to all types of the polygon. But you can derive the formula by your self.
Let us now derive the area of the regular hexagon. We need to know a few terms before starting the calculation.
Circumcircle: A circumcircle is a circle drawn outside the hexagon and it touches the vertices of the hexagon. Therefore, the radius of the circle is equal to the radius of the hexagon.
Incircle: An incircle is a circle drawn inside the hexagon and it touches the midpoints of the sides of the hexagon. The apothem is the radius of the incircle or the distance from the midpoint of the hexagon to the midpoint of the side.
Area of a hexagon
Let ‘r’ be the radius of the circumcircle, ‘a’ be the apothem (a line from the polygon’s centre to any side, this line cuts the side at the midpoint ) and ‘n’ be the length of each side.
We can see that the hexagon can be divided into the isosceles triangles (two sides of length r) i.e. the two vertices will be the vertices of the hexagon connecting to the midpoint of the hexagon. There will be a total of 6 triangles as there are six sides.
Area of the triangle = Δt = 1/2 × base × height
Here, let the base of the triangle be of size n and a height equal to apothem ‘a’. Then,
Δt = 1/2 × n × a
Let us further divide the triangle into two equal right-angled triangles of base n/2 each and a height equal to a.
The overall revolution is equal to 360o. The angle ∠ACB for the triangle will be 360o/6 = 60o. Since we have further divided the triangle. So, for each small triangle, let’s say ACD, the angle ∠ACD will be 60o/2 = 30o or π/6 radians.
We know that tangent is equal to the ratio of opposite side of angle (perpendicular) to the adjacent side (base).
tan(θ) = opposite side / adjacent side
tan(30o) = (n/2) / a
1/√3 = n / 2a
2/√3 = n / a
a / n = √3/2
a = (√3n/2)
Putting the value of a in the above equation of the area of the triangle
Δt = 1/2 × n × (√3n/2)
Δt = √3/4 × n2
We are not done yet. This is the area of one triangle. The hexagon consists of 6 triangles. Hence,
Area of hexagon = Δ = 6 × Δt
Δ = 6 × √3/4 × n2
Δ = 3√3/2 × n2
- So, formula for the area of a hexagon = 3√3/2 × n2
Where ‘n’ is the length of the side of hexagon.
Perimeter of hexagon
If n is the length of one side of the hexagon, then the sum of the lengths of all the sides i.e.
Perimeter = p = 6n
Let’s solve some questions related to the area and perimeter of hexagon.
If the length of a side of a regular hexagon is 5 cm. Find its area and perimeter.
Length of each side = n = 5 cm
The area can be found by the following formula:
Area = Δ = 3√3/2 × n2
= 3√3/2 × 52
= 3√3/2 × 25
= 64.598 cm2 or 64.6 cm2
The area of the hexagon is 64.6 cm2.
The perimeter can be calculated as follows:
p = 6n
= 6 × 5 = 30 cm
The perimeter of the hexagon is 30 cm.
If the perimeter of the regular hexagon is 36/√3 cm. Calculate its area.
Perimeter = p = 6n
p = 36/√3 cm
36 / √3 = 6n
n = 36 / 6√3
n = 6/√3 cm
The length of each side is 6/√3 cm.
Finding the area.
Δ = 3√3/2 × n2
n = 6/√3 cm
Δ = 3√3/2 × (6/√3)2
= 3√3/2 × 12
= 31.177 cm2 or 31.18 cm2
The area of the hexagon is 31.18 cm2.