A polygon is a two-dimensional (2-D) closed figure made up of straight line segments. In geometry, hexagon is a polygon with 6 sides. If the lengths of all the sides and the measurement of all the angles are equal, such hexagon is called a regular hexagon. In other words, sides of a regular hexagon are congruent.

There is a predefined set of formulas for the calculation of perimeter and area of a regular hexagon which is collectively called as hexagon formula. The hexagon formula for a hexagon with the side length of a, is given as:

**Perimeter of an Hexagon = 6a**

** Area of an Hexagon = \(\frac{3\sqrt{3}}{ 2} \times a^{2}\)**

Hexagon formula helps us to compute the area and perimeter of hexagonal objects. Honeycomb, quartz crystal, bolt head, Lug/wheel nut, Allen wrench, floor tiles etc are few things which you would find a hexagon.

**Properties of a Regular Hexagon:**

- It has six sides and six angles.
- Lengths of all the sides and the measurement of all the angles are equal.
- The total number of diagonals in a regular hexagon is 9.
- The sum of all interior angles is equal to 720 degrees, where each interior angle measures 120 degrees.
- The sum of all exterior angles is equal to 360 degrees, where each exterior angle measures 60 degrees.

**Derivation:**

Consider a regular hexagon with each side *a *units.

**Formula for area of a hexagon:** Area of a hexagon is defined as the region occupied inside the boundary of a hexagon.

In order to calculate the area of a hexagon, we divide it into small six isosceles triangles. Calculate the area of one of the triangles and then we can multiply by 6 to find the total area of the polygon.

Take one of the triangles and draw a line from the apex to the midpoint of the base to form a right angle. The base of the triangle is ** a**, the side length of the polygon. Let the length of this line be

**.**

*h*The sum of all exterior angles is equal to 360 degrees. Here, ∠AOB = 360/6 = 60°

∴ θ = 30°

We know that the tan of an angle is opposite side by adjacent side,

Therefore, \( tan\theta = \frac{\left ( a/2 \right )}{h}\)

\(tan30 = \frac{\left ( a/2 \right )}{h}\)

\(\frac{\sqrt{3}}{3}= \frac{\left ( a/2 \right )}{h}\)

\(h= \frac{a}{2}\times \frac{3}{\sqrt{3}}\)

The area of a triangle = \(\frac{1}{2}bh\)

The area of a triangle=\(\frac{1}{2}\times a\times \frac{a}{2}\times \frac{3}{\sqrt{3}}\)

=\(\frac{3}{\sqrt{3}}\frac{a^{2}}{4}\)

Area of the hexagon = 6 x Area of Triangle

Area of the hexagon = \(6\times \frac{3}{\sqrt{3}} \times \frac{a^{2}}{4}\)

**Area of an Hexagon = \(\frac{3\sqrt{3}}{ 2} \times a^{2}\)**

**Formula for perimeter of a hexagon:** Perimeter of a hexagon is defined as the length of the boundary of the hexagon. So perimeter will be the sum of the length of all sides. The formula for perimeter of a hexagon is given by:

Perimeter = length of 6 sides

**Perimeter of an Hexagon = 6a**

**Solved examples:**

**Question 1:** Calculate the area and perimeter of a regular hexagon whose side is 4.1cm.

**Solution:** Given, side of the hexagon = 4.1 cm

**Area of an Hexagon = \(\frac{3\sqrt{3}}{ 2} \times a^{2}\)**

Area of an Hexagon = **\(\frac{3\sqrt{3}}{ 2} \times 4.1^{2}\) **= 43.67cm²

Perimeter of the hexagon**= **6a**= **6 × 4.1 = 24.6cm

**Question 2: **Perimeter of a hexagonal board is 24 cm. Find the area of the board.

**Solution**: Given, perimeter of the board = 24 cm

Perimeter of an Hexagon = 6a

24 cm = 6a

a = 24/6 = 4 cm

Area of an Hexagon = **\(\frac{3\sqrt{3}}{ 2} \times 4^{2}\)**= 41.57cm²

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