# Hexagon

## Definition

A hexagon is a polygon having six straight sides and six angles. The word ‘hexagon’ came from the Greek word ‘hex’ meaning ‘six’ and ‘gonia’ meaning ‘corner, angle’.

When all the six sides and angles of a hexagon are equal, it is called a regular hexagon. Otherwise, it is an irregular hexagon. In our article, we deal with only regular hexagon to better understand the concepts regarding the shape.

## Properties

1. Has six sides of equal length; in ⎔ABCDEF, AB = BC = CD = DE = EF = FA
2. Has six interior angles, each measuring 120°; so ∠ABC =∠BCD = ∠CDE = ∠DEF = ∠EFA = ∠FAB = 120°
3. The sum of the interior angles is 720°; so ∠ABC +∠BCD + ∠CDE + ∠DEF + ∠EFA +∠FAB = 720°
4. Has six exterior angles, each measuring 60°
5. Has nine diagonals; shown as AC, AD, AE, BD, BE, BF, CE, CF, and CA

## Formulas

### Perimeter

The total distance covered around the edge of the hexagon. The formula is given below:

Find the perimeter of a regular hexagon whose each side measure 25 cm.

As we know,
Perimeter (P) = 6a,where a = 25 cm
= 6 x 25 cm
= 150 cm

### Area

The total space enclosed by the hexagon. The formula is given below:

Find the area of a regular hexagon whose each side measures 9.9 cm

As we know,
Area (A) = 33/2 x (a)2, where a = 9.9 cm
= 3√3/2 x 9.9 x 9.9
= 254.33 cm2

Problem: Finding the area of a regular hexagon when only the PERIMETER is known

Find the area of a regular hexagonal board whose perimeter is 30 cm

Given, perimeter (P) of the hexagonal board = 30 cm
As we know,
Perimeter (P) = 6a
So, 30 = 6a
a = 30/6 cm = 5 cm
As we know,
Area (A) = 33/2 x (a)2
Hence, Area (A) of the hexagonal board = 3√3/2 x 5 x 5
= 64.87 cm2

## Types of Hexagon

1. Regular Hexagon: All six sides are of equal length and all six angles are of equal measure. Each of the internal angles in a hexagon measure 120°. In a regular hexagon, the apothem is equal to the radius of the inscribed circle. Drawing diagonal lines between the nonadjacent vertices of a regular hexagon results in a perfect star shape known as hexagram. A regular hexagon has six lines of symmetry.
2. Irregular Hexagon: Does not have sides all of equal length and angles of equal measures
3. Convex Hexagon: Have all vertices pointing outwards and thus no internal angles are greater than 180°
4. Concave Hexagon: Have at least one vertex pointing inwards and thus have at least one internal angle greater than 180°

## Applications and Examples of the Shape in Real Life

• Cell of a beehive
• The outer parts of a metal nut
• Found in the football as white structures
• The compound eyes of a fly
• The other end of a pencil