What Is the Area of a Hexagon Formula for Regular and Irregular Hexagons
The concept of area of hexagon plays a key role in mathematics and is widely applicable to both real-life situations and exam scenarios. From honeycomb designs to geometry in school and competitions, mastering how to find the area of a hexagon is essential for students in Classes 6–12 and for those preparing for JEE, ICSE, CBSE, and Olympiad exams.
What Is Area of Hexagon?
A hexagon is a two-dimensional polygon with six sides and six angles. The area of a hexagon refers to the space enclosed within its six sides, measured in square units like cm² or m². If all sides and angles are equal, it is known as a regular hexagon, and the area can be found using special formulas. You’ll find this concept applied in areas such as polygons, geometry, and mathematical modeling.
Key Formula for Area of Hexagon
Here’s the standard formula for the area of a regular hexagon with side length s:
Area = (3√3/2) × s²
Alternatively, if the apothem (a) and perimeter (P) are known:
Area = (1/2) × Perimeter × Apothem = 3 × a × s
And, if you know the radius (distance from center to vertex, r):
Area = (3√3 × r²) / 2
| Given | Formula |
|---|---|
| Side only (s) | (3√3/2) × s² |
| Apothem (a) and side (s) | 3 × a × s |
| Radius (r) | (3√3/2) × r² |
Formula Derivation (Why Does It Work?)
The area of a regular hexagon can be found by dividing it into 6 equilateral triangles of side s. The area of each triangle is (√3/4) × s². So, for the hexagon:
Area = 6 × (√3/4) × s² = (3√3/2) × s²
This stepwise approach helps you remember the formula and also gives you a proof for board exams.
Step-by-Step Illustration
- Suppose the side of a regular hexagon is s = 6 cm.
Use the formula: Area = (3√3/2) × s² - Plug in the value: s² = 6 × 6 = 36
Area = (3√3/2) × 36 - Multiply: (3×36) = 108; 108 ÷ 2 = 54
Area = 54√3 cm² - Final Answer: The area of the hexagon is 54√3 cm²
Speed Trick or Vedic Shortcut
Here’s a quick shortcut: If you only know the perimeter (P) and apothem (a), use:
Area = (1/2) × P × a
This saves you time—especially in MCQs where apothem is provided. Remember, in a regular hexagon, apothem ≈ 0.866 × side, which is useful for mental math in geometry problems!
Try These Yourself
- Find the area of a regular hexagon with side 8 cm.
- Given apothem a = 5 cm, side s = 6 cm, find the area.
- If the radius of a regular hexagon is 10 cm, what is its area?
- Check: Is a figure with sides 6 cm, 6 cm, 6 cm, 8 cm, 6 cm, 6 cm regular? Can you use the simple formula?
Frequent Errors and Misunderstandings
- Using the hexagon area formula for irregular hexagons (it only works for regular hexagons).
- Forgetting to square the side in the formula.
- Using perimeter instead of side in the area formula.
- Missing the square root in √3—always check your calculations!
Relation to Other Concepts
Understanding the area of hexagon helps you solve polygon area questions, compare triangle areas (since a hexagon is made up of triangles), and tackle mensuration word problems. It’s especially helpful for properties of hexagons and for calculating the perimeter of polygons in combination problems.
Classroom Tip
A quick way to remember the area of hexagon formula is: “Three root three by two times side squared” — make a chant of it! Many Vedantu teachers use visual aids and triangle breakdowns to help students lock in the formula and avoid confusion.
We explored area of hexagon—from definition, formula, stepwise problems, and quick tricks. Keep practicing with Vedantu’s live classes and doubt sessions to become confident with all polygon area topics. Understanding this formula is the first step to cracking geometry sections in school and competitive exams!
Related Vedantu Resources
- Properties of Hexagon – Learn about side, angle, and symmetry properties.
- Area of Polygon – Discover how to find areas of different polygons.
- Area of Triangle – See how a hexagon splits into triangles for easy area calculation.
- Geometry – Find all fundamental geometry topics in one place.
FAQs on Area of Hexagon Complete Guide with Formula and Examples
1. What is the formula for the area of a hexagon?
The formula for the area of a regular hexagon is A = (3√3/2) a², where a is the length of one side.
- This formula applies only to a regular hexagon (all sides and angles equal).
- It is derived by dividing the hexagon into 6 equilateral triangles.
- Each triangle has area (√3/4)a², and multiplying by 6 gives (3√3/2)a².
2. How do you find the area of a regular hexagon step by step?
To find the area of a regular hexagon, use the formula A = (3√3/2) a² and substitute the side length.
- Step 1: Measure the side length (a).
- Step 2: Square the side length (a²).
- Step 3: Multiply by (3√3/2).
3. What is the area of a hexagon with side length 6?
The area of a regular hexagon with side length 6 is 54√3 square units.
- Use the formula: A = (3√3/2)a².
- Substitute a = 6.
- A = (3√3/2) × 36 = 54√3.
4. How do you find the area of a hexagon using the apothem?
The area of a regular hexagon using the apothem is A = (1/2) × perimeter × apothem.
- Perimeter (P) = 6a (since there are 6 equal sides).
- Apothem (r) is the distance from the center to the midpoint of a side.
- So, A = (1/2) × P × r.
5. How do you calculate the area of an irregular hexagon?
The area of an irregular hexagon is found by dividing it into triangles or rectangles and adding their areas.
- Split the hexagon into simpler shapes.
- Find the area of each part separately.
- Add all areas together.
6. Why is the formula for the area of a regular hexagon (3√3/2)a²?
The formula (3√3/2)a² comes from dividing a regular hexagon into 6 equilateral triangles.
- Each triangle has area (√3/4)a².
- Multiply by 6: 6 × (√3/4)a² = (3√3/2)a².
7. What is the difference between the area and perimeter of a hexagon?
The area of a hexagon measures the space inside it, while the perimeter measures the total length around it.
- Area is measured in square units (e.g., cm², m²).
- Perimeter is measured in linear units (e.g., cm, m).
- For a regular hexagon, perimeter = 6a.
8. Can you find the area of a hexagon if you know the perimeter?
Yes, you can find the area of a regular hexagon from the perimeter if you first calculate the side length.
- Step 1: Find side length a = perimeter ÷ 6.
- Step 2: Use A = (3√3/2)a².
9. What units are used for the area of a hexagon?
The area of a hexagon is measured in square units.
- Examples include cm², m², in², and ft².
- The unit depends on the unit used for the side length.
- If the side is in meters, the area will be in square meters (m²).
10. Where is the area of a hexagon used in real life?
The area of a hexagon is commonly used in tiling, architecture, engineering, and nature-based designs.
- Honeycombs use hexagonal shapes for maximum efficiency.
- Floor tiles and paving blocks often use hexagonal patterns.
- Mechanical parts and nuts are sometimes hexagonal in shape.
