Area of hexagon formula
A hexagon is a polygon which has six sides and six angles. The word hexagon is derived from the Greek words “Hexa” and “gonia” which means “six” and “corner” respectively. There are two types of a hexagon:
This Story also Contains
- Table of contents
- Area of regular hexagon formula
- Area of regular hexagon using apothem
- Area of irregular hexagon
- Examples
Regular hexagon: All the sides and all the angles of a regular hexagon are of equal value. The order of rotational symmetry of a regular hexagon is six. It is composed of six equilateral triangles.
Irregular hexagon: All the sides and angles of an irregular hexagon are not the same.
The area of a hexagon is the space contained inside its sides. The area of a hexagon can be calculated in a number of different ways. The various techniques primarily depend on the way of splitting of the hexagon. We shall examine numerous approaches to calculating the area of a hexagon in this article.
Table of contents
Area of regular hexagon formula
Area of regular hexagon using apothem
Area of irregular hexagon
Examples
FAQs
Area of regular hexagon formula
Area of a regular hexagon can be calculated by dividing the hexagon into six equilateral triangles. After dividing the hexagon into triangles, we compute the area of one equilateral triangle and multiply it by six to get the area of the hexagon.
Area of equilateral triangle is given by the following formula:
Area\: of \: equilateral\: triangle=\frac{\sqrt3}{4}a^2
Where, a is the length of the side of the equilateral triangle.
Since, the area of the hexagon is six times the area of the triangle we get the formula of area of hexagon as:
Area\: of\: hexagon=\frac{3\sqrt3}{2}a^2
A:The derivation involves dividing the hexagon into six equilateral triangles. The area of each triangle is (√3 * s^2) / 4, where s is the side length. Multiplying this by 6 gives the hexagon's area: (3√3 * s^2) / 2.
A:For a regular hexagon, the area (A) and perimeter (P) are related by the formula A = (P^2) / (24 * tan(30°)). This shows how increasing the perimeter affects the area, an important concept in optimization problems.
A:Using √3 keeps the answer in exact form, preserving precision for further calculations. It also maintains the connection to the hexagon's geometry, as √3 relates to its 60° angles.
A:A regular hexagon's six-fold rotational symmetry allows us to calculate its area by finding the area of one triangular section and multiplying by 6, simplifying the process and highlighting geometric symmetry's practical applications.
A:A hexagon's area is larger than a square with the same side length. The ratio is approximately 2.598:2, showing how shape affects area even with equal side lengths.
Area of regular hexagon using apothem
Area of equilateral triangle using the height of the triangle is given by:
Area\: of \: equilateral\: triangle=\frac{1}{2}ah
Where, h is the height of the equilateral triangle and a is the base of the triangle.
Since, the area of the hexagon is six times the area of the triangle we get the formula of area of hexagon as:
Area\: of\: hexagon=3ah
The perimeter of a hexagon is equal to six times the side of the hexagon. Hence, the perimeter ‘p’ is equal to 6a.
Using the perimeter of the hexagon, the area of hexagon can be calculated by the following formula:
Area\: of\: hexagon=\frac{1}{2}ph
A:The √3 comes from the trigonometry of the hexagon. It represents the ratio between the apothem (distance from center to midpoint of a side) and half the side length, which is key to calculating the area.
A:The apothem is the perpendicular distance from the center of a regular hexagon to any of its sides. It's crucial for area calculation as it forms the height of the six equilateral triangles that make up the hexagon.
A:An inscribed circle touches all six sides of the hexagon. Its radius is equal to the apothem of the hexagon, which is a key component in calculating the hexagon's area, showing the interconnection between these geometric concepts.
A:Side length (s) and radius (R) are related in a regular hexagon: R = s. The choice between formulas using s or R depends on the given information, but both lead to the same result.
A:A regular hexagon can be divided into six equilateral triangles. Its area is exactly six times the area of one of these triangles, showcasing the relationship between these shapes.
Area of irregular hexagon
There is no precise formula for determining the area of an irregular hexagon. The area of an irregular hexagon can be calculated by partitioning it into several shapes like squares, rectangles and triangles. The area of these shapes can be calculated individually and then summed up to determine the area of the hexagon.
A:The formula for a regular hexagon (all sides equal) is A = (3√3 * s^2) / 2, where s is the side length. For irregular hexagons, the area is typically calculated by dividing the shape into triangles and summing their areas.
A:A hexagon is a six-sided polygon. Understanding its area is crucial in various fields, from architecture to engineering, as it helps in calculating space, material requirements, and optimizing designs.
A:If you double the side length of a hexagon, its area increases by a factor of 4. This is because the area is proportional to the square of the side length, demonstrating the concept of scaling in geometry.
A:Hexagons are efficient because they tessellate (fit together without gaps) and provide maximum area for a given perimeter. This efficiency is seen in beehives and other natural structures, relating geometry to real-world applications.
A:Using trigonometry, we can find the area by calculating (6 * ½ * s^2 * sin(60°)), where s is the side length. This method uses the sine function to determine the height of the equilateral triangles forming the hexagon.
Examples
Example 1: Find the area of a regular hexagon with side length equal to 5m.
Answer: Given, side of hexagon (a) = 5m.
Therefore,
Area\: of\: hexagon=\frac{3\sqrt3}{2}a^2\\
=\frac{3\sqrt3}{2}5^2m^2\\
=64.95m^2
Example 2: If the length of side of a regular hexagon is equal to 4m and the length of apothem is equal to 3.464m then, find the area of the hexagon.
Answer: Given, the length of side (a) = 4m
The length of apothem (h) = 3.464m
The area of hexagon is given by:
Area\: of\: hexagon=3ah\\
=3 \times 4 \times 3.464 m^2\\
=41.568 m^2
Example 3: Find the side of the regular hexagon whose area is equal to 260 sq. units.
Answer: Given area is equal to 260 sq. units.
Let the side of the hexagon be a. Then,
Area\: of\: hexagon = \frac{3\sqrt3}{2}a^2\\
\frac{3\sqrt3}{2}a^2 = 260\\
a= 10\: units
A:Hexagon area calculations are used in various fields: in architecture for designing structures, in engineering for optimizing material use, and in nature studies for understanding honeycomb structures, showing the practical applications of this geometric concept.
A:To find the side length (s) from the area (A), use the formula s = √((2A) / (3√3)). This involves solving the area formula for s, demonstrating algebraic manipulation of geometric formulas.
A:Increasing each side by 1 unit increases the area non-linearly. The new area would be (3√3 * (s+1)^2) / 2, where s is the original side length. This demonstrates how small changes in dimensions can significantly affect area.
A:Hexagons are often used in optimization problems due to their efficiency in covering area with minimal perimeter. Understanding their area helps in solving problems like efficient packing or minimizing material use in engineering designs.
A:In crystallography, hexagonal close packing is a common atomic arrangement. Understanding hexagon area is crucial for calculating atomic densities and predicting material properties, linking geometry to material science.
