Fiona's Circle Area Calculation A Step-by-Step Solution

In this article, we will delve into the world of geometry to solve a problem involving circles. Circle area calculation is a fundamental concept in mathematics, and this exercise provides a practical application of the relevant formula. The problem states that Fiona draws a circle with a diameter of 14 meters. Our goal is to determine the area of this circle. To solve this, we will first revisit the key concepts related to circles, including the definitions of diameter, radius, and area. Understanding these foundational elements is crucial for accurately applying the formula and arriving at the correct answer. We will break down the steps involved in the calculation process, ensuring clarity and precision in our solution. By the end of this article, you'll not only understand how to solve this particular problem but also gain a deeper appreciation for the relationship between a circle's dimensions and its area. This skill is essential for various applications in fields like engineering, architecture, and even everyday problem-solving. Remember, the key to success in mathematics lies in mastering the fundamentals, and this exercise is an excellent way to strengthen your grasp of circle geometry. So, let's embark on this mathematical journey and uncover the solution to Fiona's circular challenge.

Before we jump into the calculation, it's essential to define the key terms: diameter, radius, and area, particularly in the context of a circle. The diameter of a circle is the straight-line distance that passes through the center of the circle, connecting two points on the circle's circumference. It's essentially the longest chord that can be drawn within a circle. The radius, on the other hand, is the distance from the center of the circle to any point on its circumference. Importantly, the radius is always half the length of the diameter. This relationship is crucial for our calculation, as the formula for the area of a circle uses the radius. Understanding the connection between the diameter and radius is fundamental to grasping circle geometry. Now, let's talk about area. The area of a circle is the amount of two-dimensional space enclosed within the circle's circumference. It's a measure of the surface covered by the circle. Unlike the perimeter (or circumference), which measures the distance around the circle, the area measures the space inside. The area of a circle is calculated using a specific formula that involves a mathematical constant known as pi (π). This formula is a cornerstone of circle geometry, allowing us to determine the area based on the radius (or diameter). In the next section, we'll delve into the formula itself and apply it to solve Fiona's circle problem.

The formula for calculating the area of a circle is a fundamental concept in geometry, expressed as A = πr², where A represents the area, π (pi) is a mathematical constant approximately equal to 3.14159, and r is the radius of the circle. This formula reveals a direct relationship between the radius of a circle and its area. Squaring the radius means that even a small change in the radius can significantly impact the area. The constant π (pi) is an irrational number, meaning its decimal representation goes on infinitely without repeating. It represents the ratio of a circle's circumference to its diameter and is a ubiquitous constant in mathematics and physics. Understanding the formula A = πr² is crucial for solving a wide range of problems involving circles, from simple calculations to more complex applications in engineering and design. This formula allows us to determine the area of any circle, provided we know its radius. The formula highlights the importance of the radius as the key parameter in determining a circle's area. In our problem, we are given the diameter, so we will need to first calculate the radius before applying the formula. This step-by-step approach ensures accuracy and a clear understanding of the process. Now, let's apply this formula to Fiona's circle and find the area. We will carefully substitute the values and perform the calculations to arrive at the solution. Remember, the accurate application of the formula is essential for obtaining the correct result.

Now, let's apply our knowledge to solve the problem. Fiona draws a circle with a diameter of 14 meters. The first step is to determine the radius of the circle. As we discussed earlier, the radius is half the diameter. Therefore, the radius (r) of Fiona's circle is 14 meters / 2 = 7 meters. Now that we have the radius, we can use the formula for the area of a circle, A = πr². Substituting the value of the radius (r = 7 meters) into the formula, we get A = π(7 meters)². Squaring the radius, we have 7² = 49. Therefore, the area A = 49π square meters. This is the exact area of Fiona's circle, expressed in terms of π. In many cases, it's preferred to leave the answer in this form, as it avoids rounding errors associated with approximating π. However, if we need a numerical approximation, we can substitute π with its approximate value (3.14159) and perform the multiplication. But for this problem, the answer choices are given in terms of π, so we have our solution. The area of Fiona's circle is 49π square meters. This step-by-step solution demonstrates how we can effectively use the formula for the area of a circle to solve a practical problem. By breaking down the problem into smaller, manageable steps, we can ensure accuracy and clarity in our calculations. In the next section, we will examine the answer choices and select the correct one.

Having calculated the area of Fiona's circle to be 49π square meters, the next step is to identify the corresponding answer from the given options. The answer choices are:

A. 7π m² B. 14π m² C. 28π m² D. 49π m²

By comparing our calculated area (49π m²) with the answer choices, we can clearly see that option D, 49π m², matches our result. Therefore, the correct answer is D. This process of comparing the calculated result with the given options is crucial in problem-solving. It helps ensure that we have arrived at the correct solution and haven't made any errors in our calculations. In this case, the answer choices are presented in terms of π, making it straightforward to identify the correct one. If the answer choices were numerical approximations, we might need to approximate π and compare the numerical values. However, in this scenario, the direct match makes the selection process simple and efficient. Identifying the correct answer is the final step in solving the problem, and it's important to double-check the result to ensure accuracy. Now that we have successfully calculated the area of Fiona's circle and identified the correct answer, let's summarize our findings and highlight the key takeaways from this exercise. This will reinforce our understanding of the concepts and the problem-solving process.

In conclusion, we have successfully calculated the area of Fiona's circle, which has a diameter of 14 meters. By understanding the relationship between diameter and radius, applying the formula for the area of a circle (A = πr²), and following a step-by-step approach, we arrived at the solution: 49π square meters. This exercise reinforces the importance of understanding fundamental geometric concepts and their applications. We revisited the definitions of diameter, radius, and area, and we learned how to effectively use the formula for the area of a circle. The key takeaways from this exercise include:

1. The radius of a circle is half its diameter.
2. The formula for the area of a circle is A = πr².
3. Accurate substitution and calculation are crucial for problem-solving.
4. Expressing the answer in terms of π provides the exact area.
5. Comparing the calculated result with the answer choices is essential for verification.

This problem demonstrates how mathematical concepts can be applied to real-world scenarios. Whether it's calculating the area of a circular garden, designing a round table, or solving more complex engineering problems, the principles we've discussed here are invaluable. By mastering these fundamental concepts, you'll be well-equipped to tackle a wide range of geometric challenges. Remember, practice is key to solidifying your understanding. Try solving similar problems with different dimensions to further enhance your skills. With consistent effort, you'll become more confident and proficient in applying mathematical principles to solve real-world problems. This concludes our exploration of Fiona's circle problem. We hope this step-by-step guide has been helpful and informative.