Area of Hexagon Formula - Explanation, Types, Solved Examples and FAQs

Edited By Team Careers360 | Updated on Jul 02, 2025 05:19 PM IST

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:

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.

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Table of contents
Area of regular hexagon formula
Area of regular hexagon using apothem
Area of irregular hexagon
Examples

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.

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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

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

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.

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

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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

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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

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Frequently Asked Questions (FAQs)

1. What is a polygon?

A polygon is a two-dimensional, closed geometrical shape that is flat or planar and is bordered by straight sides. Its sides are not curved or curled.

2. How many sides and angles does a hexagon have?

A hexagon is a polygon with six sides and six angles.

3. What are the types of hexagon?

A hexagon has two types: Regular hexagon and irregular hexagon.

If all the six sides and six angles of a hexagon are of equal value then the hexagon is regular hexagon otherwise, the hexagon is an irregular hexagon.

4. What is the perimeter of a hexagon?

The perimeter of a hexagon is the sum of length of all six sides of a hexagon. If one side of a regular hexagon is equal to ‘a’ then its perimeter is equal to six times the side of the hexagon that is equal to ‘6a’.

5. What is the formula for calculating the area of a regular hexagon?

The area of a regular hexagon is given by following formulas:

Area\: of\: hexagon=\frac{3\sqrt3}{2}a^2

Area\: of\: hexagon=3ah

Area\: of\: hexagon=\frac{1}{2}ph

Here, ‘a’ is the side of the hexagon.

‘h’ is the length of the apothem of the hexagon.

‘p’ is the perimeter of the hexagon.

6. How does the formula for the area of a regular hexagon differ from an irregular one?

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.

7. What is a hexagon and why is its area important?

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.

8. How does the area of a hexagon change if you double its side length?

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.

9. Why is the hexagon considered an efficient shape in nature?

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.

10. Can you explain how to calculate the area of a hexagon using trigonometry?

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.

11. Can you explain how to derive the area formula for a regular hexagon?

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.

12. What's the relationship between a hexagon's area and its perimeter?

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.

13. Why is the area of a hexagon often expressed in terms of √3 rather than a decimal approximation?

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.

14. How does the concept of symmetry play a role in hexagon area calculations?

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.

15. How does the area of a hexagon compare to that of a square with the same side length?

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.

16. Why does the formula for a regular hexagon's area involve √3?

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.

17. Can you explain the concept of apothem in relation to hexagon area?

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.

18. How does inscribing a circle in a hexagon affect area calculations?

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.

19. What's the difference between using side length and radius in hexagon area calculations?

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.

20. How does the area of a hexagon relate to the area of an equilateral triangle?

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.

21. How can you use hexagon area to solve real-world problems?

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.

22. How can you find the side length of a hexagon if you only know its area?

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.

23. How does the area of a hexagon change if you increase all its sides by 1 unit?

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.

24. How does understanding hexagon area contribute to the study of optimization problems?

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.

25. What's the significance of the hexagon in crystallography and how does its area play a role?

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.

26. How does the concept of hexagon area extend to hyperbolic geometry?

In hyperbolic geometry, the area of a hexagon is related to its angle deficit (2π minus the sum of its interior angles). This contrast with Euclidean geometry introduces students to non-Euclidean geometries and their unique properties.

27. How does the area of a hexagon change if it's projected onto a curved surface?

Projecting a hexagon onto a curved surface (like a sphere) changes its area due to distortion. This concept, important in cartography, demonstrates how geometric properties can change in different spatial contexts.

28. How does the area of a hexagon change if it's stretched in one direction?

Stretching a hexagon in one direction creates an irregular hexagon. Its area would increase, but not proportionally to the stretch. Calculating this new area would require dividing the shape into triangles or trapezoids.

29. What's the significance of the 30-60-90 triangle in hexagon area calculations?

The 30-60-90 triangle is half of an equilateral triangle, which forms 1/12 of a regular hexagon. Understanding this triangle's properties is key to deriving and understanding the hexagon area formula.

30. How can you use the area of a hexagon to find its circumradius?

If you know the area (A) of a regular hexagon, you can find its circumradius (R) using the formula R = √((2A) / (3√3)). This demonstrates the relationship between area and the circle that circumscribes the hexagon.

31. Why is the hexagon area formula not simply the square of the side length multiplied by a constant?

Unlike simpler shapes like squares, the hexagon's area involves trigonometric ratios due to its angles. The √3 in the formula reflects these ratios, showing how more complex shapes require more sophisticated area formulas.

32. How does the concept of regular polygons extend from hexagons to other shapes?

The hexagon is part of the family of regular polygons. As the number of sides increases, the polygon's area formula approaches that of a circle, demonstrating the concept of limits in geometry.

33. Can you explain how to find the area of a hexagon using calculus methods?

Using calculus, we can find the area by integrating the function of half the hexagon's width with respect to its height. This method, while more complex, connects hexagon area to advanced mathematical concepts.

34. How does the area of a hexagon relate to its internal angles?

Each internal angle of a regular hexagon is 120°. This angle determines the shape of the six equilateral triangles that make up the hexagon, directly affecting its area calculation.

35. What's the relationship between a hexagon's area and the area of its circumscribed circle?

The area of a regular hexagon is approximately 82.7% of its circumscribed circle's area. This ratio demonstrates how polygons approximate circles as the number of sides increases.

36. What's the difference between using radians and degrees in hexagon area calculations?

While both lead to the same result, using radians (π/3 instead of 60°) in trigonometric functions can simplify some calculations, especially in more advanced mathematical contexts.

37. How does the concept of tessellation relate to hexagon area?

Hexagons tessellate perfectly, meaning they can cover a plane without gaps. This property, related to their area efficiency, is why they're often used in tile designs and observed in nature, like in beehives.

38. Can you explain how to find the area of a hexagon inscribed in a circle?

For a hexagon inscribed in a circle with radius R, the area is (3√3 * R^2) / 2. This formula is derived from the relationship between the circle's radius and the hexagon's side length, which are equal in this case.

39. What's the significance of the centroid in hexagon area calculations?

The centroid is the geometric center of the hexagon. While not directly used in the area formula, understanding its location (where the three main diagonals intersect) helps in more complex calculations and in understanding the hexagon's symmetry.

40. How can you use hexagon area to introduce the concept of limits to students?

As you increase the number of sides in a regular polygon, its area approaches that of a circle. Using hexagons as a starting point, you can demonstrate how adding sides brings the shape closer to a circle, introducing the concept of limits geometrically.

41. What's the connection between hexagon area and the golden ratio?

While the golden ratio is not directly involved in hexagon area calculations, comparing the areas of nested hexagons can lead to interesting ratios, providing a gateway to discuss various mathematical constants and their occurrences in geometry.

42. How does the area of a hexagon relate to its volume if extruded into 3D?

If a hexagon is extruded into 3D to form a hexagonal prism, its area becomes the base area in the volume calculation. The volume would be the hexagon's area multiplied by the height of extrusion, connecting 2D and 3D geometry concepts.

43. Can you explain how to find the area of a hexagon using vector methods?

Using vectors, you can calculate the area by taking the cross product of two adjacent sides and multiplying the result's magnitude by 3. This method connects hexagon area to linear algebra concepts.

44. What's the relationship between a hexagon's area and its diagonal length?

In a regular hexagon, the diagonal (d) is related to the side length (s) by d = s√3. Using this, we can express the area in terms of the diagonal: A = (√3 * d^2) / 4. This shows how different measurements of a hexagon are interrelated.

45. How can understanding hexagon area help in studying other polygons?

The methods used for hexagon area (dividing into triangles, using trigonometry) can be applied to other polygons. This transferable knowledge helps in understanding area calculations for more complex shapes.

46. What role does the hexagon play in the study of tessellations and why is this significant?

Hexagons are one of only three regular polygons that can tessellate a plane by themselves (along with triangles and squares). This property, related to their area efficiency, is crucial in fields like crystallography and materials science.

47. Can you explain how hexagon area relates to the study of fractals?

Hexagonal fractals, created by repeatedly replacing hexagons with smaller hexagons, demonstrate how area changes across different scales. This introduces concepts of self-similarity and scaling in fractal geometry.

48. How can you use hexagon area to introduce the concept of dimensional analysis?

The units in hexagon area calculations (length squared) provide a good opportunity to discuss dimensional analysis. This helps students understand how units behave in calculations and the importance of consistency in physical measurements.

49. What's the connection between hexagon area and the study of honeycombs in nature?

Honeybees build hexagonal cells to maximize storage space while minimizing building material. Understanding hexagon area helps explain why this shape is optimal, connecting geometry to biological efficiency.

50. How does the concept of hexagon area extend to three-dimensional analogues like hexagonal prisms or truncated octahedra?

The area of a hexagon becomes a crucial component in calculating the surface area and volume of 3D shapes like hexagonal prisms. This extension helps bridge the gap between 2D and 3D geometry.

51. Can you explain how hexagon area relates to the study of quasicrystals?

Quasicrystals often exhibit pentagonal and decagonal symmetries, which can be studied using hexagonal grids. Understanding hexagon area aids in analyzing these complex structures, connecting basic geometry to advanced materials science.

52. How does the study of hexagon area contribute to understanding geometric probability?

Problems involving randomly placed points in hexagons introduce geometric probability concepts. The hexagon's area is crucial in calculating probabilities, linking geometry to probability theory.

53. What role does hexagon area play in the design of soccer balls and other sporting equipment?

Soccer balls typically use a combination of hexagons and pentagons. Understanding hexagon area is crucial for designers to ensure proper inflation, ball roundness, and flight characteristics, showing practical applications of geometry in sports.

54. How can the study of hexagon area lead to discussions about the nature of proof in mathematics?

Deriving the hexagon area formula provides an excellent opportunity to discuss different types of mathematical proofs (geometric, algebraic, trigonometric). This helps students understand the rigor and logic behind mathematical truths.

55. What's the significance of hexagon area in the study of cellular automata and Conway's Game of Life?

Many cellular automata, including Conway's Game of Life, use hexagonal grids. Understanding hexagon area is crucial for analyzing patterns and behaviors in these systems, connecting geometry to computational and complexity theory.