Apolygonis a two-dimensional (2-D) closed figure made up of straight line segments. In geometry, hexagon is a polygon with 6 sides. If the lengths of all the sides and the measurement of all the angles are equal, such hexagon is called a regular hexagon. In other words, sides of a regular hexagon are congruent.
There is a predefined set of formulas for the calculation of perimeter and area of a regular hexagon which is collectively called as hexagon formula. The hexagon formula for a hexagon with the side length of a, is given as:
Perimeter of an Hexagon = 6a
Area of an Hexagon =
\(\begin{array}{l}\frac{3\sqrt{3}}{ 2} \times a^{2}\end{array} \)
Hexagon formula helps us to compute the area and perimeter of hexagonal objects. Honeycomb, quartz crystal, bolt head, Lug/wheel nut, Allen wrench, floor tiles etc are few things which you would find a hexagon.
Properties of a Regular Hexagon:
- It has six sides and six angles.
- Lengths of all the sides and the measurement of all the angles are equal.
- The total number of diagonals in a regular hexagon is 9.
- The sum of all interior angles is equal to 720 degrees, where each interior angle measures 120 degrees.
- The sum of all exterior angles is equal to 360 degrees, where each exterior angle measures 60 degrees.
Derivation:
Consider a regular hexagon with each side a units.
Formula for areaof a hexagon: Areaof a hexagon is defined as the region occupied inside the boundary of a hexagon.
In order to calculate the area of a hexagon, we divide it into small six isosceles triangles. Calculate the area of one of the triangles and then we can multiply by6to find the total area of the polygon.
Take one of the triangles and draw a line from the apex to the midpoint of the base to form a right angle. The base of the triangle isa, the side length of the polygon. Let the length of this line beh.
The sum of all exterior angles is equal to 360 degrees. Here, ∠AOB = 360/6 = 60°
∴ θ = 30°
We know that the tan of an angle is opposite side by adjacent side,
Therefore, \(\begin{array}{l} tan\theta = \frac{\left ( a/2 \right )}{h}\end{array} \)
\(\begin{array}{l}tan30 = \frac{\left ( a/2 \right )}{h}\end{array} \)
\(\begin{array}{l}\frac{\sqrt{3}}{3}= \frac{\left ( a/2 \right )}{h}\end{array} \)
\(\begin{array}{l}h= \frac{a}{2}\times \frac{3}{\sqrt{3}}\end{array} \)
The area of a triangle = \(\begin{array}{l}\frac{1}{2}bh\end{array} \)
The area of a triangle= \(\begin{array}{l}\frac{1}{2}\times a\times \frac{a}{2}\times \frac{3}{\sqrt{3}}\end{array} \)
= \(\begin{array}{l}\frac{3}{\sqrt{3}}\frac{a^{2}}{4}\end{array} \)
Area of the hexagon = 6 x Area of Triangle
Area of the hexagon = \(\begin{array}{l}6\times \frac{3}{\sqrt{3}} \times \frac{a^{2}}{4}\end{array} \)
Area of an Hexagon =
\(\begin{array}{l}\frac{3\sqrt{3}}{ 2} \times a^{2}\end{array} \)
Formula for perimeterof a hexagon: Perimeter of a hexagon is defined as the length of the boundary of the hexagon. So perimeter will be the sum of the length of all sides. The formula for perimeterof a hexagon is given by:
Perimeter = length of 6 sides
Perimeter of an Hexagon = 6a
Solved examples:
Question 1: Calculate the area and perimeter of a regular hexagon whose side is 4.1cm.
Solution: Given, side of the hexagon = 4.1 cm
Area of an Hexagon =
\(\begin{array}{l}\frac{3\sqrt{3}}{ 2} \times a^{2}\end{array} \)
Area of an Hexagon =
\(\begin{array}{l}\frac{3\sqrt{3}}{ 2} \times 4.1^{2}\end{array} \)
= 43.67cm²Perimeter of the hexagon= 6a= 6 × 4.1 = 24.6cm
Question 2: Perimeter of a hexagonal board is 24 cm. Find the area of the board.
Solution: Given, perimeter of the board = 24 cm
Perimeter of an Hexagon = 6a
24 cm = 6a
a = 24/6 = 4 cm
Area of an Hexagon =
\(\begin{array}{l}\frac{3\sqrt{3}}{ 2} \times 4^{2}\end{array} \)
= 41.57cm²To solve more problems on the topic, downloadBYJU’S-The Learning App.
As a geometry enthusiast with a deep understanding of polygonal shapes and their properties, let me delve into the intricacies of hexagons and the mathematical formulations associated with them.
The article discusses hexagons, which are two-dimensional closed figures comprised of straight line segments. A hexagon, a specific type of polygon, stands out with its six sides. When these sides are of equal length and the angles are congruent, it transforms into a regular hexagon. The regular hexagon is a fascinating geometric entity, and to analyze it, we rely on the hexagon formula.
The hexagon formula provides us with the means to calculate both the perimeter and area of a regular hexagon. For a hexagon with a side length of 'a,' the formulas are as follows: [ \text{Perimeter of a Hexagon} = 6a ] [ \text{Area of a Hexagon} = \frac{3\sqrt{3}}{2} \times a^{2} ]
These formulas serve as powerful tools to compute the perimeter and area of hexagonal objects like honeycombs, quartz crystals, bolt heads, lug/wheel nuts, Allen wrenches, and floor tiles. The properties of a regular hexagon are also outlined, emphasizing its symmetry and defining characteristics:
- It has six sides and six angles.
- The lengths of all sides and the measurement of all angles are equal.
- The total number of diagonals in a regular hexagon is 9.
- The sum of all interior angles is 720 degrees, with each interior angle measuring 120 degrees.
- The sum of all exterior angles is 360 degrees, with each exterior angle measuring 60 degrees.
The article proceeds to provide a derivation for the area of a regular hexagon. By breaking down the hexagon into six isosceles triangles, the area is computed using trigonometric principles. The final formula for the area of a hexagon is derived as: [ \text{Area of a Hexagon} = \frac{3\sqrt{3}}{2} \times a^{2} ]
The perimeter is calculated straightforwardly as the sum of the lengths of all six sides: [ \text{Perimeter of a Hexagon} = 6a ]
The article concludes with solved examples, illustrating how to apply these formulas in practical situations. These examples involve calculating the area and perimeter of regular hexagons given specific side lengths. This comprehensive coverage equips readers with the knowledge and tools needed to understand and work with regular hexagons in various contexts.
