Calculating The Area Of Fiona's Circle A Step By Step Guide

When tackling geometry problems, it's essential to break down the information provided and apply the relevant formulas. In this case, we're presented with a scenario where Fiona draws a circle, and we need to determine its area. The key piece of information we have is the diameter of the circle, which is given as 14 meters. From this, we can find the radius, and then finally, calculate the area using the appropriate formula. This problem perfectly exemplifies how understanding basic geometric concepts and formulas can help solve practical problems.

To accurately calculate the area of Fiona's circle, it’s important to understand the relationship between the diameter, radius, and area of a circle. The diameter is the distance across the circle passing through the center, while the radius is the distance from the center to any point on the circle's edge. The radius is always half the length of the diameter. This fundamental relationship is crucial for solving various circle-related problems.

In this particular problem, we're given that Fiona’s circle has a diameter of 14 meters. This is our starting point. To find the area, we need the radius first. As mentioned, the radius is half the diameter. So, in this case, the radius of Fiona’s circle is 14 meters / 2 = 7 meters. Keeping track of the units (meters in this case) is important to ensure our final answer is also in the correct units (square meters for area).

Once we have the radius, we can move on to the formula for the area of a circle. The area of a circle is given by the formula: Area = πr², where π (pi) is a mathematical constant approximately equal to 3.14159, and r is the radius of the circle. This formula is a cornerstone of geometry and is widely used in various applications, from calculating the surface area of cylindrical objects to determining the space covered by circular gardens.

Understanding this formula is crucial for solving this problem and many other similar problems. It directly links the radius of a circle to its area, allowing us to calculate the area if we know the radius, and vice versa. It’s also worth noting that the area is proportional to the square of the radius, which means that if you double the radius, you quadruple the area. This relationship highlights the significance of the radius in determining the size of a circle.

Now that we know the radius of Fiona's circle is 7 meters, we can calculate the area using the formula Area = πr². This is where the direct application of the formula comes into play. We substitute the value of the radius (r = 7 meters) into the formula and perform the calculation.

Substituting r = 7 meters into the formula, we get:

Area = π * (7 meters)²

First, we need to square the radius, which means multiplying it by itself:

(7 meters)² = 7 meters * 7 meters = 49 square meters

So now our equation looks like this:

Area = π * 49 square meters

This can be written more simply as:

Area = 49π square meters

This is the exact area of Fiona's circle, expressed in terms of π. It's important to note that we leave the answer in terms of π unless we are specifically asked to approximate it using a numerical value for π (like 3.14 or 3.14159). Keeping the answer in terms of π provides the most accurate representation of the area.

Looking at the answer choices, we can see that the options are also given in terms of π. This confirms that we've followed the correct approach by leaving π in our answer. The final step is to match our calculated area (49π square meters) with the given options.

After calculating the area of Fiona's circle, we found it to be 49π square meters. Now, we need to compare this result with the given answer choices to identify the correct one. This step is crucial to ensure we select the right option and demonstrate our understanding of the problem-solving process.

The answer choices provided are:

A. 7π m²

B. 14π m²

C. 28π m²

D. 49π m²

By directly comparing our calculated area (49π square meters) with these options, it becomes clear that option D, 49π m², matches our result perfectly. This indicates that we have correctly applied the formula for the area of a circle and accurately performed the calculations.

The other options can be easily eliminated because they do not correspond to the area we calculated. Option A (7π m²) is significantly smaller, while options B (14π m²) and C (28π m²) are also incorrect. This process of elimination reinforces our confidence in the correctness of option D.

Therefore, the correct answer is D. 49π m². This signifies that the area enclosed by Fiona's circle, which has a diameter of 14 meters, is exactly 49π square meters. This answer is consistent with the principles of geometry and demonstrates a clear understanding of how to calculate the area of a circle given its diameter.

To further solidify our understanding, let's examine why the other answer choices are incorrect. This will not only confirm our correct answer but also deepen our comprehension of the underlying concepts.

• Option A: 7π m²

  This option is incorrect because it seems to be calculating an area using the radius divided by 2 (7 meters / 2 = 3.5 meters) instead of using the actual radius (7 meters) in the area formula. It might also represent a misunderstanding of the relationship between the radius and the area, possibly confusing it with a different formula or concept. This highlights the importance of correctly identifying the radius and applying the area formula (πr²) accurately.

• Option B: 14π m²

  This option is incorrect because it appears to be using the diameter (14 meters) directly in the area calculation, without first finding the radius. This demonstrates a fundamental misunderstanding of the formula for the area of a circle, which requires the radius, not the diameter. The diameter needs to be halved to obtain the radius before being used in the formula. This mistake emphasizes the necessity of remembering and applying the correct formula and the relationship between diameter and radius.

• Option C: 28π m²

  This option is incorrect as it doesn't directly relate to a simple misapplication of the formula. It's possible that this answer results from a more complex error, such as multiplying the diameter by π or making a mistake in squaring the radius. The appearance of 28 doesn't directly stem from a clear single error in applying the area formula, suggesting a more convoluted misunderstanding or calculation mistake. This highlights the significance of carefully reviewing each step in the calculation to prevent errors.

By understanding why these options are incorrect, we reinforce our understanding of the correct method and the underlying principles of calculating the area of a circle. This deeper comprehension helps prevent similar errors in future problems.

Understanding the area of a circle isn't just an abstract mathematical concept; it has numerous practical applications in the real world. From engineering and architecture to everyday tasks, the ability to calculate circle area is essential.

In engineering and construction, calculating the area of circles is crucial for designing circular structures, pipes, and other components. For example, engineers need to know the cross-sectional area of pipes to determine fluid flow rates. Architects use circle area calculations when designing circular buildings, domes, or arches. These calculations ensure structural integrity and efficient use of materials.

In everyday life, we encounter circles in countless situations. Think about pizzas, circular gardens, or even the base of a cylindrical can. Calculating the area of these circles helps us determine the amount of material needed, the space they occupy, or the quantity of something they can hold. For example, if you're planning to build a circular patio, you'll need to calculate the area to determine how much paving material to purchase. Similarly, knowing the area of a pizza helps you estimate how many slices you can get.

Technology also relies heavily on circle area calculations. In computer graphics, circles and circular shapes are fundamental elements. Calculating their areas is essential for rendering images, creating animations, and designing user interfaces. In telecommunications, the coverage area of a cell tower is often modeled as a circle, and understanding its area helps optimize network planning.

Even in agriculture, the area of circular fields is an important consideration for irrigation and crop yield estimation. Farmers use these calculations to determine the amount of water and fertilizer needed, as well as to predict the potential harvest.

These examples illustrate the wide range of applications for circle area calculations. Mastering this concept provides a valuable tool for solving problems in various fields and understanding the world around us.

In conclusion, the problem of finding the area of Fiona's circle provides a clear example of how geometric principles are applied. By understanding the relationship between the diameter, radius, and area of a circle, we can accurately solve this type of problem. The key steps involved identifying the radius (half of the diameter), applying the formula Area = πr², and performing the calculations carefully.

We successfully calculated the area of Fiona's circle to be 49π square meters, which corresponds to answer choice D. This answer is obtained by correctly substituting the radius (7 meters) into the area formula and simplifying the expression. The other answer choices were incorrect due to misapplication of the formula or misunderstanding of the relationship between diameter and radius.

Furthermore, we explored the real-world applications of circle area calculations, highlighting their importance in various fields such as engineering, architecture, everyday life, technology, and agriculture. This demonstrates that understanding circle area is not just a theoretical exercise but a practical skill with wide-ranging uses.

By mastering the concepts and techniques presented in this problem, you can confidently tackle similar geometry challenges and appreciate the relevance of mathematics in the world around us. The ability to calculate the area of a circle is a valuable tool that will serve you well in various academic and practical contexts.