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 x²

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.

height² = a² – (a/2)² (from pythagoras theorem)

             = √3a²/2

Area of triangle  = 1/2 * a * √3a/2

                            = √3a²/4

Area of hexagon = 6 * Area of  triangle

                             = 6 * √3/4 * x²

                             = 3√3/2 * x²

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 * x²

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

                                              = 6 * √3/4 * x²

                                              = 3√3/2 * x²

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 a²

= 3√3/2 (3 * 3)

                             = 27 √3/2 cm²

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 a²

                             = 3√3/2(2√3 * 2√3)

                             = 18√3 cm²

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 a²

                             = 3√3/2(6 * 6)

                             = 54 √3 cm²

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 a²

                            = 3√3/2(2√6 * 2√6)

                            = 36√3 cm²