Calculating The Area Of Fiona's Circle A Step-by-Step Solution

This article will explore how to calculate the area of Fiona's circle, a common mathematical problem involving circles and their properties. Understanding the relationship between a circle's diameter, radius, and area is crucial for solving this type of problem. We'll break down the steps involved, ensuring a clear and comprehensive understanding of the solution.

Understanding the Problem

The core of the problem lies in determining the area of a circle given its diameter. Fiona draws a circle with a diameter of 14 meters, and our task is to find the area of this circle. To solve this, we need to recall the formula for the area of a circle and how it relates to the circle's radius. The radius, being half the diameter, is a critical component in this calculation. This problem underscores the importance of understanding fundamental geometric concepts and their practical applications. We will first identify the given information, then determine the formula for the area of a circle, and finally substitute the known values to calculate the area. This step-by-step approach will help clarify the solution process.

Firstly, recognizing the importance of the diameter, it is crucial to know that the diameter is the distance across the circle passing through the center. In this case, the diameter is given as 14 meters. The next step is to find the radius, which is half the diameter. Dividing the diameter by 2, we find the radius to be 7 meters. Understanding this relationship between diameter and radius is foundational to solving circle-related problems. The area of a circle is the space enclosed within its boundary. Calculating this area requires a specific formula that uses the radius as its primary input. This highlights the interconnectedness of various geometrical properties and the importance of grasping these relationships for effective problem-solving.

Understanding the formula for the area of a circle, which is 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, is vital. This formula is a cornerstone of geometry and is used extensively in various mathematical and scientific applications. It is essential to remember this formula and understand the significance of each component. The radius, as we've already established, is half the diameter, and pi is a constant that represents the ratio of a circle's circumference to its diameter. By squaring the radius and multiplying it by pi, we can accurately determine the area of the circle. This formula provides a concise and effective method for calculating the area, making it a fundamental tool in geometric problem-solving.

Calculating the Area

The actual calculation involves substituting the known values into the formula. We know the radius (r) is 7 meters. Substituting this into the area formula (A = πr²), we get A = π * (7 m)². This means we need to square the radius (7 meters) and then multiply the result by π. Squaring 7 meters gives us 49 square meters. Therefore, the area of Fiona's circle is 49π square meters. This straightforward calculation demonstrates the application of the formula and the importance of understanding the units involved. The area is expressed in square meters because it represents a two-dimensional space. This step-by-step process ensures accuracy and clarity in the solution.

Breaking down the calculation further, squaring 7 meters results in 49 square meters. Then, multiplying this by π gives us the area in terms of π. Since π is an irrational number, leaving the answer in terms of π is often the most precise way to represent the area. This approach avoids rounding errors that can occur when using a decimal approximation of π. The final answer, 49π square meters, is therefore the exact area of Fiona's circle. This method emphasizes the importance of mathematical precision and the use of constants in formulas. Understanding how to manipulate these values and apply them correctly is crucial for accurate calculations.

The final answer is 49π m², which corresponds to option D. This means the area of Fiona's circle is exactly 49 times the value of pi in square meters. This answer provides a clear and concise solution to the problem. The process of arriving at this answer involved understanding the relationship between diameter and radius, recalling the area formula, and substituting the known values correctly. This comprehensive approach highlights the importance of a solid understanding of geometric principles and their application in practical problems. The ability to solve such problems is crucial for various applications in mathematics, science, and engineering.

Why Other Options Are Incorrect

To further solidify our understanding, it's beneficial to examine why the other options are incorrect. This helps reinforce the correct solution and clarifies any potential misconceptions about circle area calculations. By analyzing common errors, we can better understand the nuances of the problem and avoid similar mistakes in the future. This process of elimination and error analysis is a valuable tool in problem-solving.

Option A, 7π m², is incorrect because it seems to be calculating the area using the radius divided by 2 instead of squaring it. This is a common mistake where the formula is not applied correctly. The area formula requires squaring the radius, which means multiplying the radius by itself. Dividing the radius by 2 and then multiplying by π will not yield the correct area. This error highlights the importance of accurately remembering and applying the formula. The correct application of the formula is crucial for obtaining the correct solution.

Option B, 14π m², is incorrect as it might be confusing the diameter with the radius in the area formula. This option appears to have multiplied the diameter (14 meters) by π directly, without first calculating the radius and then squaring it. This mistake underscores the importance of distinguishing between the diameter and the radius and understanding their roles in the area calculation. Confusing these two values can lead to significant errors in the final result. The radius is the critical value to use in the area formula, not the diameter.

Option C, 28π m², is also incorrect. This result may come from multiplying the diameter (14 meters) by the radius (7 meters) and then multiplying by π. This calculation does not correspond to the correct formula for the area of a circle. It's important to remember that the area formula involves squaring the radius, not multiplying the diameter by the radius. This type of error highlights the need for a clear understanding of the formula and its components. The correct formula, A = πr², must be applied to accurately calculate the area.

Key Takeaways

In summary, the area of Fiona's circle is 49π square meters. This result was obtained by correctly identifying the radius from the given diameter, recalling the area formula (A = πr²), and substituting the values appropriately. The problem highlights the fundamental concepts of geometry, including the relationship between diameter and radius, and the application of the area formula. By understanding these concepts and practicing problem-solving techniques, we can confidently tackle similar mathematical challenges.

The key takeaways from this problem include the importance of accurately applying formulas, understanding the relationship between different geometric properties (such as diameter and radius), and avoiding common mistakes in calculations. The ability to solve such problems is essential for various applications in mathematics, science, and engineering. This understanding not only helps in academic settings but also in practical real-world scenarios where geometric calculations are required. A solid grasp of these concepts provides a strong foundation for further mathematical studies.

Furthermore, this exercise reinforces the value of step-by-step problem-solving. By breaking down the problem into smaller, manageable steps, we can minimize errors and ensure a clear and logical approach. This systematic method involves identifying the known information, determining the appropriate formula, substituting the values, and performing the calculations accurately. This approach is applicable not only to mathematical problems but also to various other problem-solving situations in different fields. The ability to approach problems methodically is a valuable skill in any discipline.

Conclusion

In conclusion, calculating the area of a circle given its diameter is a fundamental mathematical skill. Fiona's circle problem provides a practical example of how to apply geometric principles and formulas to solve real-world problems. By understanding the concepts, practicing the calculations, and avoiding common errors, we can confidently determine the area of any circle given its diameter or radius. This knowledge is essential for various applications in mathematics, science, and engineering, and it forms a strong foundation for further studies in these fields. The ability to solve such problems accurately and efficiently is a valuable asset in both academic and professional settings.