Equilateral Triangle Area Calculation Step-by-Step Guide

In geometry, understanding how to calculate the area of various shapes is fundamental. Today, we'll delve into the specifics of finding the area of an equilateral triangle given its apothem and perimeter. This comprehensive guide will walk you through the process step-by-step, ensuring you grasp the underlying concepts and can confidently tackle similar problems. Let's explore the intricacies of equilateral triangles and their area calculations.

Understanding the Properties of Equilateral Triangles

Before diving into the calculations, it's crucial to understand the unique properties of equilateral triangles. An equilateral triangle is a triangle with all three sides of equal length and all three angles equal to 60 degrees. This symmetry simplifies many calculations related to these triangles. Key properties include:

• Equal Sides: All three sides have the same length.
• Equal Angles: Each interior angle measures 60 degrees.
• Apothem: The apothem is the line segment from the center of the triangle to the midpoint of a side. It is also the radius of the inscribed circle.
• Perimeter: The perimeter is the total length of all three sides.
• Area: The area can be calculated using various methods, including using the side length, apothem, or height.

Knowing these properties is essential for solving problems related to equilateral triangles. For instance, the relationship between the apothem and the side length is crucial for our calculations today. Understanding that the apothem is also the radius of the inscribed circle helps visualize the triangle's geometry and aids in problem-solving. Furthermore, the 60-degree angles allow us to use trigonometric relationships if necessary, providing an alternative approach to finding the area.

Problem Statement: Finding the Area

Our specific problem involves an equilateral triangle with an apothem measuring 2.16 cm and a perimeter of 22.45 cm. The goal is to calculate the area of this triangle, rounded to the nearest tenth. This problem combines the concepts of perimeter, apothem, and area, requiring us to use the given information effectively to arrive at the solution. We will break down the problem into manageable steps, making the process clear and easy to follow. The ability to solve this type of problem demonstrates a solid understanding of geometric principles and problem-solving skills.

Step 1: Determine the Side Length

The perimeter of a triangle is the sum of the lengths of its three sides. Since we are dealing with an equilateral triangle, all three sides are equal in length. Therefore, we can find the length of one side by dividing the perimeter by 3. In this case, the perimeter is 22.45 cm. So, the side length (s) can be calculated as:

s = Perimeter / 3
s = 22.45 cm / 3
s ≈ 7.4833 cm

We now know that each side of the equilateral triangle is approximately 7.4833 cm long. This is a crucial piece of information, as the side length is directly related to the area and other properties of the triangle. The precision of this calculation is important, as rounding too early can lead to inaccuracies in the final answer. Therefore, we will keep the value as precise as possible until the final rounding step. This meticulous approach ensures that our final result is as accurate as possible, reflecting a strong understanding of mathematical precision.

Step 2: Relate Apothem to Side Length

The apothem of an equilateral triangle is related to its side length. The apothem (a) is the distance from the center of the triangle to the midpoint of a side. In an equilateral triangle, the apothem, half of a side, and a line segment from the center to a vertex form a 30-60-90 right triangle. The relationship between the apothem and the side length (s) in an equilateral triangle is:

a = (s * √3) / 6

However, in this problem, we are given the apothem and need to use it to verify our calculations and find the area. The above formula is derived from the properties of 30-60-90 triangles, which are fundamental in geometry. The understanding of this relationship is not only useful for this problem but also for various other geometric problems involving equilateral triangles and hexagons. This connection between the apothem and side length allows us to move seamlessly between different parameters of the triangle, making complex calculations more manageable. Moreover, it highlights the interconnectedness of geometric concepts, reinforcing the importance of a holistic understanding of the subject.

Step 3: Calculate the Area Using Apothem and Perimeter

The area (A) of an equilateral triangle can be calculated using the apothem (a) and the perimeter (P) with the following formula:

A = (1/2) * a * P

This formula is derived from the more general formula for the area of a regular polygon, where the area is half the product of the apothem and the perimeter. Plugging in the given values, we have:

A = (1/2) * 2.16 cm * 22.45 cm
A = 24.246 cm²

This calculation directly links the apothem and perimeter to the area, demonstrating the efficiency of this formula. The formula is particularly useful when the apothem and perimeter are known, as it provides a straightforward method for finding the area without needing to calculate the side length explicitly. Furthermore, this approach underscores the elegance of geometric formulas in providing concise solutions to complex problems. The ability to apply this formula correctly is a testament to a strong grasp of geometric principles and their practical applications.

Step 4: Round to the Nearest Tenth

The final step is to round the calculated area to the nearest tenth. The calculated area is 24.246 cm². Rounding this to the nearest tenth gives us:

Area ≈ 24.2 cm²

Therefore, the area of the equilateral triangle, rounded to the nearest tenth, is 24.2 cm². This rounding step is crucial for providing the answer in the desired format, as specified in the problem statement. It also highlights the importance of understanding rounding rules and their application in mathematical problems. Presenting the answer in the correct format demonstrates attention to detail and a thorough understanding of the problem's requirements. Moreover, it ensures that the solution is both accurate and clearly communicated.

Alternative Method: Calculating the Area Using Side Length

Alternatively, the area of an equilateral triangle can be calculated using the side length (s) with the following formula:

A = (√3 / 4) * s²

Using the side length we calculated earlier (s ≈ 7.4833 cm), we can find the area:

A = (√3 / 4) * (7.4833 cm)²
A ≈ (√3 / 4) * 55.9998 cm²
A ≈ 0.433 * 55.9998 cm²
A ≈ 24.2479 cm²

Rounding this to the nearest tenth also gives us approximately 24.2 cm². This alternative method provides a way to verify our previous calculation and reinforces the relationship between the side length and the area of an equilateral triangle. The consistency between the two methods strengthens our confidence in the accuracy of the solution. Moreover, it illustrates that different approaches can lead to the same result in mathematics, highlighting the versatility of geometric principles.

Conclusion

In conclusion, the area of the equilateral triangle with an apothem of 2.16 cm and a perimeter of 22.45 cm is approximately 24.2 cm², rounded to the nearest tenth. We arrived at this solution by first determining the side length from the perimeter, then using the apothem and perimeter to calculate the area. We also verified our result using the formula that directly relates the side length to the area, ensuring accuracy and consistency. This problem demonstrates the importance of understanding the properties of equilateral triangles and applying the appropriate formulas to solve geometric problems. By breaking down the problem into manageable steps and using clear, logical reasoning, we successfully found the solution. This approach can be applied to a wide range of geometric problems, reinforcing the value of a systematic problem-solving strategy.

The correct answer is D. 24.2 cm². This comprehensive guide not only provides the solution but also explains the underlying concepts and methodologies, empowering you to tackle similar problems with confidence. Understanding the relationships between different parameters of geometric shapes is crucial for mastering geometry, and this example serves as a valuable learning tool.