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A polygon which has 6 number of sides is known as a Hexagon. The word ‘hexa’ means ‘six’ and ‘gon’ means ‘angle’. Since the polygon has 6 sides which consequently forms 6 angles, hence it is known as a Hexagon.

A polygon which has 6 number of sides (or edges) and 6 number of angles is a Hexagon. As shown in the figure on the left, hexagons have 6 vertices (or corners), 6 edges (or sides) and 6 angles.

Based on the measurements of the sides, hexagons are classified into 2 types: Regular Hexagons and Irregular Hexagons.

1)

a) Regular Hexagons has 6 equal sides and 6 equal interior angles.

b) As a hexagon has even number of sides, hence the opposite sides of a regular hexagon are parallel to each other.

c) A line drawn from the center of the regular hexagon to any of the vertices will have the same length as the side length, as shown in the figure below.

d) All the regular hexagons are convex, which means that all its 6 vertices point outward.

e) The line segment joining any two non-adjacent vertices in a polygon is known as a ‘Diagonal’. The diagonals of a regular hexagon divide the hexagon into 6 equilateral triangles as shown in the figure on the right.

a) Irregular Hexagons do not have 6 equal sides or 6 equal interior angles.

b) The opposite sides may or may not be parallel to each other.

c) An irregular hexagon can be convex in shape or concave in shape. A Convex polygon is a polygon which has all the vertices pointing outward. But in a concave polygon, one or more vertices point inward towards the center of the polygon. Because of this reason, in a concave polygon one or more interior angles is greater than 180°.

d) A line drawn through a concave hexagon (depending on where the line is drawn) can intersect the hexagon at more than 2 points. The below figure shows the line intersecting the hexagon at 4 points.

e) In a concave hexagon, all the diagonal do not lie inside the hexagon. One or more diagonals lie outside the hexagon also, as shown in the figure below.

The sum of all the interior angles of any regular polygon can be calculated using the formula given below:

Since a hexagon has 6 sides, hence n = 6.

Now Sum, S = (6 – 2) * 180° = 720°

Therefore,

The measure of each interior angle of any regular polygon can be calculated using the formula given below:

Since a hexagon has 6 sides, hence n = 6.

So, Each Interior angle = (6 – 2) / 6 * 180°

The measure of each exterior angle of any regular polygon can be calculated using the formula given below:

Each Exterior angle of Regular Convex Hexagon = 360°/6

Since a hexagon has 6 sides, hence n = 6.

Therefore, number of diagonals in a hexagon = 6 * (6 – 3)/2 =

Perimeter is the total length calculated when all the side lengths of the polygon are combined together. Perimeter of a regular or irregular polygon can be calculated by adding all the side lengths of the polygon.

Therefore Perimeter of a Regular Hexagon of side length ‘s’ (as shown in the figure on the right) will be written as, P = s + s + s + s + s + s

Perimeter of a Regular Hexagon, P = 6 * s ==> Perimeter, P = 6 * 7m = 42m

Given the side lengths of the hexagon in the figure.

Perimeter of a Hexagon = Sum of all the side lengths.

Therefore, Perimeter, P = 4m + 7m + 3m + 2m + 8m + 2m = 26m

As mentioned above, diagonal of a regular hexagon divide the hexagon into 6 equal triangles, also known as 6 equilateral triangles. So if we find the area of one equilateral triangle, then the area of all the 6 triangles will be known, and then the area of the hexagon will be the triangle areas added together.

Given a regular hexagon as shown in the figure above, where point ‘C’ is the center of the hexagon.

Triangle CPQ is an equilateral triangle, as all the angles inside triangle CPQ are equal to 60° (half of the interior angle 120°). Hence all its sides are also equal.

Therefore, let the side lengths of CP = PQ = CQ = s

CM is the perpendicular drawn to the side PQ. Let CM = h

As ‘M’ becomes the midpoint of side PQ, hence MQ = s/2 (half of the side length of PQ).

Now in triangle CMQ, we can apply the Pythagorean Theorem to get the relationship between the height ‘h’ of the triangle, and the side length ‘s’.

Hence, h

So, h

Now, Area of triangle CPQ = 1/2 * base * height.

This implies, Area A = 1/2 * s * h ==> A = 1/2 * s * (s * √3/2) ==> A = s

Therefore, Area of triangle CPQ = s

Now, a regular hexagon consists of 6 such congruent equilateral triangles.

Hence, Area of a Regular Hexagon = 6 * s

Given that the side length, s = 5m

Area of a regular hexagon, A = 3/2 * s

Hence, Area = 3/2 * 5

Since an irregular hexagon does not have equal sides or equal angles, hence we cannot use the method or formula of the regular hexagon. For an irregular hexagon, we can calculate area by using various methods. Let us look at an example below:

In the given figure, we observe that the side lengths are given and the lengths of the diagonals are also given.

We can see that the irregular hexagon is split into 4 triangles A, B, C and D.

Since the side lengths of each triangle are given, we can use Heron’s formula.

Triangle A:

s = (5 + 4 + 7)/2 ==> s = 8

Now Area of triangle A = √[s(s-a)(s-b)(s-c)] = √[(8* (8 - 5) * (8 – 4) * (8 - 7)]

Area of Triangle A = √(8 * 3 * 4 * 1)

Triangle B:

s = (7 + 7 + 6)/2 = 10

Area of triangle B = √[(10 * (10 - 7) * (10 - 7) * (10 - 6)] = √(10 * 3 * 3 * 4)

Similarly using Heron’s Formula as shown above, we get the areas of triangles C and D as well.

Area of Triangle C

Now, Area of the Irregular Hexagon = Area of Triangle A + Area of Triangle B + Area of Triangle C + Area of Triangle D

Area of the Hexagon = 9.8m

A Hexagonal Tessellation is a tessellation formed when hexagons are arranged on a plane as shown in the figure below. This pattern for a Regular Hexagonal Tessellation is identical. In the figure below, we can see the vertex marked. At each vertex, we can observe that 3 hexagons are meeting. Since each hexagon has 6 sides, hence this kind of tessellation is named as 6.6.6 Tessellation.

As shown in the figure below, a hexagonal prism has 2 hexagonal bases and 6 rectangular faces.

Lateral area of the hexagonal prism is the sum of the areas of the 6 rectangular faces.

If the height of the prism is ‘h’ and the side of the base regular hexagon is ‘s’, then:

Area of each rectangular face = s * h

Therefore,

The perpendicular from the center of the hexagon to its base side is also known as the ‘Apothem’ (shown as ‘d’ in the figure below).

If the base side length is ‘s’, then as mentioned above Apothem or height of the hexagon is

Then, Area of the Hexagon = 3 * s * d = 3 * s * √3/2 * s = 3/2 * √3 * s

Since there are 2 such base hexagons, hence Bases Area = 2 * 3/2 * √3 * s

Surface Area = Bases Area + Lateral Area

Hence,

Given that the side length of the base regular hexagon, s = 4 inches

Height of the prism, h = 6 inches

Surface Area of a Regular Hexagonal Prism = (3√3 * s

Therefore, Surface Area = (3√3 * 4

Volume of a Prism is the amount of space occupied within the boundaries or edges of the prism.

Volume of a Hexagonal Prism = Area of the Base * Height of the Prism

Therefore,

Given that the side length of the base regular hexagon, s = 4 inches

Height of the prism, h = 6 inches

Volume of a Hexagonal Prism = 3/2 *√3 * s

Hence, Volume = 3/2* √3 * 4