What Makes a Hexagon Unique? Key Characteristics and Formulas
A polygon is a geometrical shape enclosed with straight lines. It is a two-dimensional shape and a common example of a polygon is a square or a rectangle. Moreover, polygons can be regular and irregular in shape. A regular polygon is a polygon that has equal sides as well as equal angles. While an irregular polygon is a polygon that does not have equal sides and therefore, the angles are also not equal.
Geometry is one of the most interesting sections of geometry, the craze of getting your own geometry box to perform little experiments on the paper. We at Vedantu believe in keeping that excitement and curiosity intact within every child. In this article, we shall be discussing one such important topic of geometry that is the properties of Hexagon.
Table of Content:
Properties of Hexagon - an introduction
What are the Properties of Hexagon?
Special Properties of Hexagon
Frequently asked questions
Hexagon Properties Geometry
A hexagon is a six-sided polygon with the sum of internal angles as 720o. If you look at a hexagon, you can see that it consists of triangles. Some day to day objects that are hexagonal in shape are a honeycomb, floor tiles, bolt head, etc. A regular hexagon has equal angles and equal sides. Moreover, there are two types of the hexagon- convex hexagon and concave hexagon.
A convex hexagon is similar to a regular hexagon as it has interior angles lesser than 180o.
While a concave hexagon is opposite to convex as it has one or more than one angle greater than 180o.
The area of a regular hexagon can be calculated using a formula developed by theorists. The formula is given below:
A =√(332)
s2 ; where
A - is the area of the hexagon
s - is the length of each side
Moreover, the perimeter for a regular hexagon can also be calculated by a formula that is mentioned below:
Perimeter = 6s; where
s - is the length of each side
Moving forward there are some special properties of the hexagon that are unique to this polygon and they shall be discussed further in this article
Special Properties of Hexagon
The first and most important property of a regular hexagon is that it has six sides. Hexa means six and therefore a hexagon is supposed to have six side.
The next property is in continuation of the first property. As there are six sides, there are also six angles in a hexagon.
The lengths of each side of a regular hexagon are equal.
The angles of each side of a regular hexagon are equal.
A regular hexagon has nine diagonals.
Since all angles are equal in a regular hexagon, each angle is 120o and the summation of all the interior angles is 720o.
In terms of exterior angles, all the angles are again equal in a regular hexagon. The summation of all exterior angles is 360o with each angle equal to 60o.
A regular hexagon also has six axes of symmetry. Three of these axes pass through diagonals opposite to the vertices and the remaining three pass through the middles of opposite edges.
When a straight line is drawn from the center of a regular hexagon that joins one of its vertices, it can lead to the formation of six identical triangles.
The central angle of each of these triangles is 60o. Also, the remaining two angles are also 60o. Therefore, it can be concluded that these are equilateral triangles. A regular hexagon is formed of 6 equilateral triangles that are identical in terms of lengths and angles of each side.
Another important property of a hexagon is circumcircle. A regular polygon has six vertices which can form a circle connecting these vertices. This circle is called a circumscribed circle or circumcircle of the polygon. The center of this circle is the same as the center of the polygon. Similarly, the radius of the circumcircle is the distance from the center to the vertex of the polygon. This radius is referred to as circumradius. Likewise, the diameters of the circumcircle are the same as the diameter of the hexagon.
Apart from the circumcircle, there is another circle that can be formed in a hexagon. This circle is called an incircle. As the name suggests, this circle is inscribed in the hexagon and is formed through the middle of the hexagon sides. Similar to circumradius, the radius of the incircle is called inradius that is formed by lines connecting the center of the hexagon with the sides or vertex. The center for the circle is the same as the center of the hexagon. Moreover, the incircle is tangent to all six sides of the hexagon.
There are various properties of a hexagon. Some of the major properties of a regular hexagon are its equal sides, equal angles, nine diagonals, circumcircle, and incircle. Moreover, it is necessary to know about the angles, interior, and exterior of a regular hexagon. Further applications of these properties have been observed in trigonometry, coordinate geometry, and also when calculating the area of a hexagon or parts of a hexagon.
FAQs on Properties of Hexagon: All You Need to Know
1. What defines a shape as a hexagon, and what are its basic properties?
A hexagon is a polygon—a two-dimensional closed shape made of straight lines. Specifically, it is defined by having six sides, six angles, and six vertices (corners). A key property of any hexagon, regular or irregular, is that the sum of its interior angles is always 720°.
2. What is the main difference between a regular and an irregular hexagon?
The primary difference lies in the uniformity of their sides and angles. A regular hexagon has all six sides of equal length and all six interior angles are equal, each measuring 120°. In contrast, an irregular hexagon has sides and/or angles of different measurements, giving it a non-uniform shape.
3. What are the key angle properties of a regular hexagon?
In a regular hexagon, the angles have specific, consistent properties based on its symmetrical shape.
- Interior Angles: Each of the six interior angles measures exactly 120°.
- Sum of Interior Angles: The total sum of all interior angles is always 720°.
- Exterior Angles: Each exterior angle measures 60°, as an interior angle and its corresponding exterior angle must sum to 180°.
4. How do you calculate the area and perimeter of a regular hexagon?
You can calculate the area and perimeter of a regular hexagon using simple formulas based on the length of one of its sides (let's call it 's'):
- Perimeter (P): The perimeter is the total length around the shape. The formula is P = 6 × s.
- Area (A): The area can be found using the formula A = (3√3 / 2) × s². This formula is derived from the total area of the six equilateral triangles that form the hexagon.
5. How many diagonals does a hexagon have, and are they all the same length?
A hexagon has a total of nine diagonals. However, they are not all equal in length. In a regular hexagon, the diagonals come in two different lengths: six shorter diagonals that connect vertices by skipping one vertex, and three longer diagonals that pass through the hexagon's centre, connecting opposite vertices.
6. Why is a regular hexagon composed of six equilateral triangles?
A regular hexagon can be divided into six identical triangles by drawing its three longest diagonals, which all intersect at the hexagon's centre. Each of these triangles is equilateral. This is because the 120° interior angles of the hexagon are bisected by the diagonals, creating 60° angles within the triangles. Since all three angles in each triangle are 60°, they are, by definition, equilateral.
7. How can you find the area of an irregular hexagon?
There is no single formula for an irregular hexagon's area because its dimensions are not uniform. The standard method is to divide the irregular hexagon into smaller, familiar shapes. You can break it down into a combination of triangles and quadrilaterals, calculate the area of each individual shape, and then sum up their areas to find the total area of the hexagon.
8. What are some real-life examples of hexagons?
Hexagons are frequently found in both nature and man-made structures due to their strength and efficiency. Some common examples include:
- A single cell of a honeycomb.
- The shape of many nuts and bolts.
- Floor tiles and paving stones.
- The hexagonal crystal structure of snowflakes.
- The basalt columns at the Giant's Causeway.
9. A hexagon is a 2D shape, so how can it have a volume?
This is a key conceptual point. A hexagon itself, being a two-dimensional (2D) shape, only has an area and does not have volume. When a question refers to the 'volume of a hexagon', it almost always means the volume of a hexagonal prism, which is a three-dimensional (3D) object that has a hexagonal base and a specific height.
10. Why are shapes like honeycombs and nuts often hexagonal in design?
The hexagonal shape is favoured in nature and engineering for its remarkable efficiency and strength. Hexagons can fit together perfectly without any gaps, a property known as tessellation. This allows bees to build honeycombs that store the maximum amount of honey using the minimum amount of wax. For nuts and bolts, the flat sides provide multiple contact points for tools like wrenches, ensuring a secure grip for tightening or loosening.
