How to Calculate the Area of a Hexagon with Side, Apothem, or Radius
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 – Formula, Examples & Stepwise Solutions
1. What is the formula for the area of a regular hexagon?
The area (A) of a regular hexagon with side length 's' is given by the formula: A = (3√3/2) × s². This formula is derived by dividing the hexagon into six equilateral triangles.
2. How do you calculate the area of a hexagon using its apothem?
The area of a regular hexagon can also be calculated using its apothem (a), which is the distance from the center to the midpoint of a side, and its perimeter (P). The formula is: A = (1/2) × a × P. Since the perimeter of a regular hexagon is 6 times its side length, you can also express this as A = 3as.
3. How is the area of a hexagon related to the area of an equilateral triangle?
A regular hexagon can be divided into six congruent equilateral triangles. Therefore, the area of a regular hexagon is six times the area of one of these equilateral triangles. The area of an equilateral triangle with side 's' is (√3/4)s².
4. Can I calculate the area of a hexagon if only the perimeter is known?
No, you cannot calculate the area of a hexagon knowing only its perimeter. You need additional information, such as the apothem or side length, to use the appropriate area formula. The perimeter alone only gives you the total length of the sides.
5. Is the area formula the same for all hexagons?
No, the formula A = (3√3/2) × s² applies only to regular hexagons (those with all sides and angles equal). For irregular hexagons, the area calculation is more complex and may involve dividing the hexagon into smaller shapes with known area formulas.
6. What are some common mistakes students make when calculating the area of a hexagon?
Common mistakes include: using the wrong formula (e.g., applying the area formula for a square or other polygon); miscalculating the side length or apothem; forgetting to square the side length in the formula; and failing to understand the distinction between regular and irregular hexagons.
7. How can I derive the area formula for a regular hexagon?
The derivation involves dividing the hexagon into six equilateral triangles. Find the area of one equilateral triangle (using the formula (√3/4)s²), and then multiply by six to get the total area of the hexagon.
8. Are there alternative methods for finding the area of a hexagon besides using formulas?
Yes, you could use a grid method to approximate the area. This involves overlaying a grid of unit squares over the hexagon and counting the squares, estimating partial squares. This method is less accurate than using the formulas.
9. What real-world applications use hexagon area calculations?
Hexagon area calculations are used in various fields, including architecture (honeycomb structures), engineering (tile design), and even biology (studying hexagonal cells).
10. How does the area of a hexagon change if you double its side length?
Doubling the side length of a regular hexagon increases its area by a factor of four (because the area formula contains the side length squared). This is because the area is directly proportional to the square of its side length.
11. Can you explain the relationship between the radius and the area of a regular hexagon?
The area of a regular hexagon can be expressed in terms of its radius (r), which is the distance from the center to a vertex: A = (3√3/2)r². Note that this is related to the side length by the fact that the side of an equilateral triangle making up the hexagon is equal to the radius.
12. How do I determine the side length of a regular hexagon if I only know its area?
Rearrange the area formula A = (3√3/2)s² to solve for 's': s = √[(2A)/(3√3)]. Substitute the known area (A) into this equation to find the side length (s).
