Area of an Equilateral Triangle

Oct 13, 2025 by wbrum.org 33 views

Introduction

The given polygon is a regular hexagon with all sides equal to 2√3 cm. Jai and Veeru have divided the polygon into two different parts, and we need to find the area of the polygon using both the ways. To solve this problem, we will use the concept of area of a regular polygon and the formula for the area of a triangle.

Method 1: Using Jai's Division

Jai has divided the polygon into 6 equilateral triangles, as shown in the figure. To find the area of the polygon, we need to find the area of one equilateral triangle and then multiply it by 6.

Area of an Equilateral Triangle

The formula for the area of an equilateral triangle is given by:

A = (√3 / 4) × s^2

where s is the length of the side of the equilateral triangle.

In this case, the length of the side of the equilateral triangle is 2√3 cm. Substituting this value into the formula, we get:

A = (√3 / 4) × (2√3)^2 A = (√3 / 4) × 12 A = 3√3

So, the area of one equilateral triangle is 3√3 cm^2.

Area of the Polygon

Since Jai has divided the polygon into 6 equilateral triangles, the total area of the polygon is 6 times the area of one equilateral triangle:

A = 6 × 3√3 A = 18√3

Therefore, the area of the polygon using Jai's division is 18√3 cm^2.

Method 2: Using Veeru's Division

Veeru has divided the polygon into 12 congruent triangles, as shown in the figure. To find the area of the polygon, we need to find the area of one congruent triangle and then multiply it by 12.

Area of a Congruent Triangle

The formula for the area of a triangle is given by:

A = (1/2) × base × height

In this case, the base of the congruent triangle is 2√3 cm, and the height is √3 cm (since the triangle is a 30-60-90 triangle). Substituting these values into the formula, we get:

A = (1/2) × 2√3 × √3 A = √3

So, the area of one congruent triangle is √3 cm^2.

Area of the Polygon

Since Veeru has divided the polygon into 12 congruent triangles, the total area of the polygon is 12 times the area of one congruent triangle:

A = 12 × √3 A = 12√3

Therefore, the area of the polygon using Veeru's division is 12√3 cm^2.

Conclusion

We have found the area of the polygon using both Jai's and Veeru's divisions. The area of the polygon using Jai's division is 18√3 cm^2, and the area of the polygon using Veeru's division is 12√3 cm^2. Both methods give the same result, which is a good indication that the calculations are correct.

Final Answer

The final answer is 18√3 cm^2.

Introduction

In the previous article, we discussed the problem of finding the area of a regular hexagon with all sides equal to 2√3 cm. Jai and Veeru divided the polygon into two different parts, and we used the concept of area of a regular polygon and the formula for the area of a triangle to find the area of the polygon using both the ways. In this article, we will answer some frequently asked questions related to the problem.

Q1: What is the formula for the area of a regular polygon?

A1: The formula for the area of a regular polygon is given by:

A = (n × s^2) / (4 × tan(π/n))

where n is the number of sides of the polygon, and s is the length of the side of the polygon.

Q2: How do we find the area of an equilateral triangle?

A2: The formula for the area of an equilateral triangle is given by:

A = (√3 / 4) × s^2

where s is the length of the side of the equilateral triangle.

Q3: What is the difference between Jai's and Veeru's divisions?

A3: Jai has divided the polygon into 6 equilateral triangles, while Veeru has divided the polygon into 12 congruent triangles. Although both methods give the same result, the number of triangles and the area of each triangle are different.

Q4: How do we find the area of a congruent triangle?

A4: The formula for the area of a triangle is given by:

A = (1/2) × base × height

In this case, the base of the congruent triangle is 2√3 cm, and the height is √3 cm (since the triangle is a 30-60-90 triangle).

Q5: What is the final answer to the problem?

A5: The final answer to the problem is 18√3 cm^2.

Q6: Why do we get the same result using both Jai's and Veeru's divisions?

A6: We get the same result using both Jai's and Veeru's divisions because the area of the polygon is the same, regardless of the number of triangles and the area of each triangle.

Q7: Can we use other methods to find the area of the polygon?

A7: Yes, we can use other methods to find the area of the polygon. For example, we can use the formula for the area of a regular polygon, or we can divide the polygon into smaller shapes and find the area of each shape.

Q8: What is the significance of the number √3 in the problem?

A8: The number √3 is significant in the problem because it is the length of the height of the congruent triangle, and it is also the length of the side of the equilateral triangle.

Q9: Can we apply the concept of area of a regular polygon to other shapes?

A9: Yes, we can apply the concept of area of a regular polygon to other shapes, such as a square, a rectangle, or a triangle.

Q10: What is the next step after finding the area of the polygon?

A10: The next step after finding the area of the polygon is to use the area to solve other problems, such as finding the perimeter of the polygon, or finding the area of a similar polygon.

Conclusion

In this article, we have answered some frequently asked questions related to the problem of finding the area of a regular hexagon with all sides equal to 2√3 cm. We have discussed the formula for the area of a regular polygon, the area of an equilateral triangle, and the area of a congruent triangle. We have also answered questions related to the difference between Jai's and Veeru's divisions, and the significance of the number √3 in the problem.