Hexagon Shape

A hexagon is an important shape in geometry. It has six sides and six angles. There are two hexagons: regular and irregular. It is important to understand hexagonal shapes to have better knowledge of mathematics. They are also important for real-life uses.

1.0Regular and Irregular Hexagon

There are great differences between the regular and irregular hexagons. 

Understanding the difference between regular and irregular hexagons is important when applying geometry to real-world design or mathematical problems. 

2.0Properties of Hexagon

The properties of hexagons differ depending on whether the hexagon is regular or irregular. 

General Properties

• It has 6 sides and 6 angles.
• The sum of the interior angles of a hexagon is 720°.
• The number of diagonals in a hexagon is 9 (calculated using the formula: $\frac{n(n-3)}{2}$, where n = 6).
• It can be divided into 4 triangles using diagonals from a single vertex.

Properties of a Regular Hexagon

• Each interior angle is exactly 120°.
• Each exterior angle is 60°.
• The hexagon can be inscribed in a circle.
• Opposite sides are parallel, and it has rotational symmetry of order 6.

3.0Interior Angles of a Hexagon

A central concept in polygon geometry is the measurement of interior angles. The interior angles of hexagons always add up to a specific total.

Formula to Calculate the Sum of Interior Angles:

$\text{ Sum of interior angles: }(n-2)\times 180^{\circ }$

Where n = number of sides

For a hexagon (n = 6):

(6 — 2) x 180° = 720°

Interior Angle of a Regular Hexagon

Since all angles in a regular hexagon are equal:

$\text{ Each interior angle }=\frac{720^{\circ }}{6}=120^{\circ }$

4.0Hexagon Area Formula

The hexagon area formula depends on whether the hexagon is regular or irregular. Let’s start with regular hexagons. 

Area of Regular Hexagon

A regular hexagon can be divided into six equilateral triangles. Using this, the area formula becomes:

$\text{ Area }=\frac{3\sqrt{3}}{2}\times a^2$

Where a is the hexagon side length.

Alternatively, you can calculate the area using the apothem (inradius) r:

$\text{ Area }=\frac{1}{2}\times \text{ perimeter }\times \text{ apothem }=\frac{1}{2}\times 6a\times r$

Area of Irregular Hexagon

For irregular hexagons, you cannot use a direct formula. Instead, divide the shape into triangles and calculate the area of each using trigonometry or coordinate geometry, then sum them.

5.0Hexagon Side Length

In practical applications, determining the hexagon side length is often necessary when designing symmetrical patterns, such as tiling or paving.

Finding Side Length from Area (Regular Hexagon)

If the area is known:

Finding Side Length from Radius

If a regular hexagon is inscribed in a circle:

Side length = radius

Using Apothem

If apothem r is known:

$\text{ Side length }=\frac{2r}{\sqrt{3}}$

6.0Examples of Hexagon in Real Life

The hexagon shape appears frequently in nature. Here are some examples of hexagon in real life: 

• Honeycomb: Bees use hexagons to build honeycombs.
• Snowflakes: Most snowflakes exhibit a six-fold symmetry due to the molecular structure of ice.
• Turtle Shells: Some species of turtles have hexagonal plates on their shells for strength and flexibility.
• Tiles and Floor Patterns: Hexagonal tiles are common in modern and classic interior designs.
• Chemical Structures: Organic compounds like benzene are represented with hexagonal diagrams in chemistry.
• Wrenches: Tools such as Allen keys and bolts often have hexagonal heads for better torque distribution.

7.0Mathematical Applications of Hexagons

The hexagon has rich mathematical properties and applications:

• Tiling Theory: A regular hexagon can tile a plane without gaps or overlaps, making it a popular shape in mathematics and architecture.
• Graph Theory: Hexagonal grids are used in modeling networks, especially in mobile communication.
• Coordinate Geometry: Regular hexagons can be represented using Cartesian coordinates for digital graphics and modelling.

8.0Conclusion

Hexagons are a masterpiece of nature and geometry. Hexagons provide strength, symmetry, and efficiency. Whether you're a student, architect, or mathematician, the hexagon offers something beautiful and functional.