Area of Hexagon

The area of a hexagon is the area of all the six (Hexa) sides of a polygon. The word Hexagon is derived from the Greek word “ Hexa” which means “ six” and “gona'' which means “corner”. If the hexagon is regular, then the length of all its 6 sides will be the same as well as all the angles will be equal. In this article, we will discuss the area of the hexagon formula and the area of hexagon formula derivation.

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What is The Area of Hexagon?

The area of a hexagon is the total area that is within all its six sides. A hexagon is a two-dimensional polygon with six sides. A hexagon has nine diagonals, and the sum of its interior angles is equal to ${720}^{\circ }$. As hexagon is a two-dimensional shape, the area of the hexagon also lies in the two-dimensional plane. The unit of area of the hexagon is determined in square units such as m², cm², in², or ft², etc.

In the figure given below, the grey space is the area of the hexagon.

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Area of Hexagon Formula (Using Side Length)

There are many different types of hexagons. The most common type of hexagon is a regular hexagon, which has sides of equal length and angles of equal measure.

The area (A) of a regular hexagon can be found easily with its side length, i.e. “s” .

Therefore, the area of the hexagon formula is given as ${\displaystyle \frac{3\sqrt{3}}{2}}{s}^2$

Area of Hexagon Formula (Using Apothem Length And Perimeter)

If we know the apothem length and perimeter of a regular hexagon, then we can use the apothem and length to find the area of the hexagon. In such case, the area of the hexagon formula is given as:

A = ${\displaystyle \frac{1}{2}}$ Pa

Here, A is the area, a is the apothem length, and P is the perimeter of the hexagon.

Area of Hexagon Formula Derivation

Divide the regular hexagon into six equilateral triangles by drawing the line segments that connect from the centre of the hexagon to each of its opposite vertices.

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Each triangle has side length s and height of ${\displaystyle \frac{\sqrt{3}}{2}}s$. Therefore, area A of one of an equilateral triangle, drawn in blue above, can be found out using:

$A={\displaystyle \frac{1}{2}}bh={\displaystyle \frac{1}{2}}s\times {\displaystyle \frac{\sqrt{3}}{2}}s={\displaystyle \frac{\sqrt{3}}{4}}{s}^2$

As there are six equilateral triangles, the area of a regular hexagon is calculated using the following formula:

Area of Hexagon Using Grid

The area of the hexagon using a grid can be calculated by determining how many square units it takes to cover its surface. Below is the unit square with a side length of 1 cm.

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A grid of unit squares can be used while calculating the area of a hexagon.

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The above grid contains unit squares that have an area of 1 cm² each. The regular hexagon given on the left side has 6 complete squares and 10 partial squares. Therefore, it has an area of approximately:

$6+{\displaystyle \frac{10}{2}}=11c{m}^2$

The regular hexagon given on the right side has 17 complete squares and 10 partial squares. Therefore, it has an area of approximately:

$17+{\displaystyle \frac{10}{2}}=22c{m}^2$

This method is not only limited to regular hexagons but can be used to find the area of any shape. However, it is only the approximate value of the area of the hexagon. More the smaller the unit square used, the higher the accuracy of the approximation. Calculating the area of a hexagon using a grid made up of 1mm² is 10 times more accurate than using a grid made up of 1 cm².

Area of Hexagon Solved Example

Question 1: What would be the area of a regular hexagon whose side is measured as 10 cm?

Answer:

Given,

The length of the side of a regular hexagon = 10 cm.

Area of regular hexagon = ${\displaystyle \frac{3\sqrt{3}}{2}}{s}^2$

By substituting the value of s = 10 cm, we get

${\displaystyle \frac{3\sqrt{3}}{2}}{10}^2$ = ${\displaystyle \frac{3\sqrt{3}}{2}}10\times 10$=${\displaystyle \frac{3\sqrt{3}}{2}}100$=$150\sqrt{3}$

Therefore, the area of a regular hexagon whose length measures 10 cm is $150\sqrt{3}c{m}^2$.

Question 2: What would be the length of the sides of the regular hexagon whose area is $600\sqrt{3}$ units?

Answer:

Given,

The area of regular hexagon =$600\sqrt{3}$ units.

Hence, ${\displaystyle \frac{3\sqrt{3}}{2}}{s}^2$ = $600\sqrt{3}$

$s^2=600\sqrt{3}\times {\displaystyle \frac{2}{3\sqrt{3}}}$

$s^2=400$

s = 20

Therefore, the length of a side of the regular hexagon is 20 units.

Practice Questions MCQs

1. What is the number of sides in a Hexagon?

1. 5

2. 6

3. 7

4. 8

Ans: B) 6

2. Regular hexagons have an interior angle of _______ degrees.

1. 108

2. 100

3. 120

4. 240

Ans: C)

Conclusion

The hexagon has six sides in mathematics; what makes this specific shape so fascinating is because it best fills a plane with equal size units and leaves no wasted space. Because of its 120-degree angles, hexagonal packing also reduces the perimeter for a given area.

The honeycomb, the geometrically ideal home for bees and their honey is the most evident natural hexagon. Bees all throughout the globe are quite skilled at creating consistent hexagons. But they are also imprinted with the shape: the hexagon is inscribed into their eyes.