How to Find the Area and Perimeter of a Regular Hexagon?
The concept of regular hexagon plays a key role in mathematics and is widely applicable in geometry, problem solving, and real-world situations such as tiling, patterns, and engineering. Regular hexagons are especially important for students in exams and maths competitions due to their symmetry and useful formulas.
What Is a Regular Hexagon?
A regular hexagon is a six-sided polygon where all the sides are equal in length and all the internal angles are the same—each measuring 120°. You’ll find this concept applied in geometry, coordinate geometry, and even nature (like in honeycombs). Regular hexagons are a special type of polygon with high symmetry and unique properties among 2D shapes.
Key Formula for Regular Hexagon
Here’s the standard formula for the area and perimeter of a regular hexagon with side length \( a \):
- Area: \( \text{Area} = \frac{3\sqrt{3}}{2} a^2 \)
- Perimeter: \( \text{Perimeter} = 6a \)
- Number of Diagonals: \( \text{Diagonals} = \frac{6(6-3)}{2} = 9 \)
You can find more geometry shape formulas in Vedantu’s formula sheet.
Properties of Regular Hexagon
| Property | Value |
|---|---|
| Number of sides | 6 (equal) |
| Internal angle | 120° each |
| Sum of internal angles | 720° |
| Lines of symmetry | 6 |
| Diagonals | 9 |
| Tessellation | Yes (tiles plane without gaps) |
Difference Between Regular and Irregular Hexagons
| Regular Hexagon | Irregular Hexagon |
|---|---|
| All sides & angles are equal | Sides or angles differ in length/degree |
| Area formula: \( \frac{3\sqrt{3}}{2} a^2 \) | Area found by dividing into triangles etc. |
| 6 lines of symmetry | May have 0, 1, or more lines of symmetry |
| Tessellates perfectly | Usually does not tessellate |
How to Draw a Regular Hexagon
- Draw a circle using a compass; mark the center as O.
- Place the compass pointer at a point A on the circle and without changing the radius, step around the circle marking points B, C, D, E, and F.
- Join each consecutive point: AB, BC, CD, DE, EF, and FA. This forms a regular hexagon.
Each side equals the radius of the circle. This construction is standard for geometry classes and projects.
Solved Example: Area of a Regular Hexagon
Question: Find the area of a regular hexagon with side length 5 cm.
Solution:
1. Area formula = \( \frac{3\sqrt{3}}{2} a^2 \ )2. Substitute \( a = 5 \) cm: \( \frac{3\sqrt{3}}{2} \times 5^2 \ )
3. Calculate: \( \frac{3\sqrt{3}}{2} \times 25 \ )
4. \( = \frac{75\sqrt{3}}{2} \) cm²
5. If \( \sqrt{3} \approx 1.73 \), then \( \frac{75 \times 1.73}{2} = \frac{129.75}{2} = 64.88 \) cm²
6. Final Answer: Area ≈ 64.9 cm²
Step-by-Step: Find Perimeter and Diagonals
Question: Find the perimeter and number of diagonals of a regular hexagon with side 7 m.
Solution:
1. Perimeter = 6 × 7 = 42 m2. Number of diagonals = \( \frac{6(6-3)}{2} = 9 \)
3. Final Answers: Perimeter = 42 m; Diagonals = 9
Speed Trick for Regular Hexagon Area
Use this Vedic shortcut: The area of a regular hexagon = ( 2.598 ) × side² (using approximate value for \( \frac{3\sqrt{3}}{2} \)). For quick math, just multiply side × side × 2.6!
Example: Side = 10 cm → 10 × 10 × 2.6 = 260 cm² (quick estimate).
Such tricks help in MCQs and competitions. Vedantu’s live classes offer more time-saving hacks for geometry.
Common Errors with Regular Hexagon
- Mixing up formulas with other polygons (like pentagon or square).
- Using wrong side length for area (not squared), or not multiplying by 2.6 in quick tricks.
- Forgetting all internal angles are equal.
Real-Life Uses of Regular Hexagon
Regular hexagons are seen in nature and daily life—beehive cells, some floor tiles, nuts and bolts, snowflakes, and even football patterns. Their symmetry makes them ideal for strong, space-filling designs. To learn about how shapes tessellate, see our tessellation guide.
Connection to Other Maths Topics
Understanding the regular hexagon helps you master advanced concepts like regular polygons, symmetry, and triangle properties (a hexagon is made of six equilateral triangles). It’s closely linked to area formulas, coordinate geometry, and trigonometry.
Classroom Memory Tip
A memorable way: “Hexagon” = “six angles, six sides, all same,” and its area formula starts with a 3 and root 3 for quick recall. Vedantu teachers teach such mnemonics with stories and patterns kids remember!
Try These Yourself
- Calculate the perimeter of a regular hexagon of side 8 cm.
- Find the area of a regular hexagon with side 4 m (use shortcut!).
- Draw and label all diagonals of a regular hexagon.
- List 3 places you see hexagons in real life.
We explored regular hexagon—from definition, properties, formula, examples, and memory tricks to real-world uses. Keep practicing with Vedantu’s maths resources and live classes—soon, hexagons will be your shortcut to scoring high in Maths!
FAQs on Regular Hexagon: Definition, Properties & Formulas
1. What is a regular hexagon in Maths?
A regular hexagon is a two-dimensional geometric shape with six equal sides and six equal angles. Each interior angle measures 120°, and the sum of all interior angles is 720°. It's a type of polygon, specifically a regular polygon because all its sides and angles are congruent.
2. What is the formula for the area of a regular hexagon?
The area (A) of a regular hexagon with side length 's' is calculated using the formula: A = (3√3/2) * s². This formula is derived by dividing the hexagon into six equilateral triangles.
3. How many lines of symmetry does a regular hexagon have?
A regular hexagon has six lines of symmetry. Three lines connect opposite vertices, and three lines bisect opposite sides.
4. What is the sum of the interior angles of a hexagon?
The sum of the interior angles of any hexagon (regular or irregular) is always 720°. This can be calculated using the formula (n-2) * 180°, where 'n' is the number of sides (6 in this case).
5. What’s the difference between a regular and irregular hexagon?
A regular hexagon has all sides and angles equal. An irregular hexagon has at least one unequal side and/or angle. The sum of interior angles remains 720° for both types.
6. What is the perimeter of a regular hexagon?
The perimeter (P) of a regular hexagon with side length 's' is simply P = 6s. This is because a regular hexagon has six equal sides.
7. How do you draw a regular hexagon using a compass and straightedge?
1. Draw a circle with your compass.
2. Without changing the compass radius, place the compass point on the circle's circumference and mark a point.
3. Repeat step 2 five more times, always keeping the same compass radius, to create six equally spaced points around the circle.
4. Connect the six points with a straightedge to form the hexagon.
8. How many diagonals does a regular hexagon have?
A regular hexagon has nine diagonals. This can be calculated using the formula n(n-3)/2, where 'n' is the number of sides (6 in this case).
9. What are some real-life examples of hexagons?
Hexagons are found in nature and design. Examples include honeycomb cells, the shape of some nuts and bolts, and certain tile patterns.
10. How is the area formula for a regular hexagon derived?
The area formula is derived by dividing a regular hexagon into six congruent equilateral triangles. The area of one equilateral triangle with side 's' is (√3/4)s². Multiplying by six (the number of triangles) gives the hexagon's area: (3√3/2)s².
11. What are the properties of a regular hexagon related to its symmetry?
A regular hexagon exhibits rotational symmetry of order 6 (it can be rotated 60° and still look the same) and reflectional symmetry with 6 lines of symmetry. This symmetry contributes to its use in tessellations.
12. Can a regular hexagon be inscribed in a circle? How?
Yes, a regular hexagon can be inscribed in a circle. The method involves using a compass to draw a circle, then using the compass radius to mark six equally spaced points on the circle's circumference, connecting these points to form the hexagon. Each side of the hexagon is equal to the radius of the circle.
