# How to Calculate the Area of a Hexagon?

A hexagon is a closed two-dimension geometrical shape. It has six sides of the same or different lengths, six vertices, and six interior angles. In the regular hexagon, all the interior angles are of 120° and exterior angles are of 60°. A regular hexagon has 9 diagonals and can be split into six equilateral triangles. Hexagon is of four types:

**Regular Hexagon:**The hexagon having all sides of equal length and with an interior angle 120° is known as a regular hexagon. The diagonals of regular hexagons are equal. They intersect at the center of the hexagon.**Irregular Hexagon:**The hexagon with unequal sides is said to be an irregular hexagon. In an irregular hexagon, all the internal angles are not equal to 120° but their sum is equal to 720°.**Convex Hexagon:**The hexagon having all the vertices pointed outward is known as a convex hexagon. It can be a regular or irregular hexagon and the interior angles are less than 180°.**Concave Hexagon:**The hexagon in which at least one vertex points inward is known as the Concave Hexagon. In this, at least one interior angle is greater than 180°.

### Area of a Hexagon

As we know the hexagon is a geometrical shape and has six sides and vertices. So the area of a hexagon(regular hexagon) is

Area of Hexagon = 3√3/2 x

^{2}

Here, x is known as the length of its sides. Here we use a regular hexagon so the length of all the sides is equal.

**Derivation for the formula:**

As we know a hexagon contains six triangles with a center point as a common vertex. Its area can be found by considering the six times the area of an equilateral triangle.

So the area of hexagon = 6 * Area of triangle

As we know the area of triangle = 1/2 * base * height

where, the base is going to be the one of the side of triangle i.e., a (let)

Now we calculate height using pythagoras theorem

From the above fig. the vertical line drawn from one of the vertex to the center of base gives the height.

height^{2 }= a^{2 }– (a/2)^{2 }(from pythagoras theorem)

= √3a^{2}/2

Area of triangle = 1/2 * a * √3a/2

= √3a^{2}/4

Area of hexagon = 6 * Area of triangle

= 6 * √3/4 * x^{2}

=** 3√3/2 * x ^{2}**

One can consider, the area of triangle = 1/2 * a * b * sin ∅ ,

where a is the side of the triangle, b is the side of the triangle and ∅ included angle between the two sides a and b

In equilateral triangle, a = b = x (let) and ∅ = 60°^{ }

So the area of equilateral triangle = 1/2 * x * x * sin 60°

= 1/2* x ^{2 }* √3/2 [ sin 60° = √3/2 ]

= √3/4 * x^{2}

Hence the area of hexagon = 6 * Area of Equilateral triangle

= 6 * √3/4 * x^{2}

= **3√3/2 * x ^{2}**

There isn’t a peculiar method to calculate the area of a regular hexagon. It is calculated by breaking down the regular hexagon into triangles and quadrilaterals and summing up their individual areas at last.

### Area of the hexagon with apothem

We can also calculate the area of a hexagon with apothem. Apothem is a line segment that is drawn from the center and perpendicular to the sides of the hexagon. So the area is

Area of Hexagon = 1/2 x perimeter(hexagon) x apothem

Or we can say

Area of hexagon = 1/2 x 6y x a = 3ya

Here, a is known as apothem and y is known as the length of the sides

### Sample Question

**Question 1. Find the area of a hexagon whose side length is 3cm.**

**Solution:**

Given the side of hexagon as 3cm i.e., a = 3cm

Area of hexagon = 3√3/2 a

^{2}

^{ }= 3√3/2 (3 * 3)= 27 √3/2 cm

^{2}

**Question 2. Find the area of the hexagon with a** **side length of** **2√3 cm.**

**Solution:**

Given the side of hexagon as 3cm i.e., a = 3cm

Area of hexagon = 3√3/2 a

^{2}= 3√3/2(2√3 * 2√3)

= 18√3 cm

^{2}

**Question 3. Find the area of a hexagon whose side length is 6cm.**

**Solution:**

Given the side of hexagon as 3cm i.e., a = 6cm

Area of hexagon = 3√3/2 a

^{2}= 3√3/2(6 * 6)

= 54 √3 cm

^{2}

**Question 4. Find the area of the hexagon with a side length of 2√6 cm.**

**Solution:**

Given the side of hexagon as 3cm i.e., a = 2√6cm

Area of hexagon = 3√3/2 a

^{2}= 3√3/2(2√6 * 2√6)

= 36√3 cm

^{2}