Calculating The Area Of A Compact Disc Step-by-Step Guide
Understanding the Problem
To effectively solve this problem, it's essential to first break down the information provided. We know the distance from the center of the compact disc to its edge is 6 centimeters. This distance represents the radius of the circular disc. The question asks us to determine the area of the compact disc, which is the total surface enclosed within the circular boundary. This is where the formula for the area of a circle comes into play. Before diving into the calculations, let's understand the formula and why it works. The formula for the area of a circle 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. This formula is derived from fundamental geometric principles and is a cornerstone of circular calculations. Understanding the formula is the first step towards solving the problem. It's not just about plugging in numbers; it's about grasping the relationship between the radius and the area of a circle. This understanding will be crucial in applying the formula correctly and interpreting the results. The radius, being the distance from the center to the edge, is the key dimension in determining the size of the circle and, consequently, its area. Now that we have a clear understanding of the problem and the formula we need, we can proceed to the next step: applying the formula with the given information.
Applying the Formula
Now that we have identified the radius (r = 6 centimeters) and understand the formula for the area of a circle (A = πr²), we can proceed with the calculation. This step involves substituting the known value of the radius into the formula and performing the necessary mathematical operations. This is a crucial step where accuracy is key. A slight error in calculation can lead to a significant difference in the final answer. Carefully substitute the radius value into the formula. In our case, we replace 'r' with 6 centimeters. This gives us A = π(6 cm)². Next, we need to square the radius. Squaring 6 centimeters gives us 36 square centimeters. So, our equation now looks like this: A = π(36 cm²). Now, we multiply 36 square centimeters by π (pi). As mentioned earlier, π is approximately equal to 3.14159. Therefore, we multiply 36 by 3.14159 to get the numerical value of the area. This multiplication is a straightforward arithmetic operation, but it's important to be precise to ensure an accurate result. The result of this multiplication will give us the area of the compact disc in square centimeters. This is the final numerical answer to our problem. However, it's not enough to just get the numerical answer. We also need to interpret the result and understand what it means in the context of the problem. The area represents the total surface enclosed within the circular boundary of the compact disc. In other words, it's the amount of space the disc covers. This understanding is crucial in various applications, such as calculating the amount of material needed to manufacture the disc or determining the storage capacity based on surface area. Now that we have performed the calculation, we can move on to the next step: interpreting the result and selecting the correct answer from the given options.
Calculating the Area and Selecting the Correct Answer
Following the formula A = πr², we substitute the radius (r = 6 cm) into the equation: A = π(6 cm)². Squaring the radius gives us A = π(36 cm²). Using the approximation π ≈ 3.14, we multiply 36 by 3.14, which results in approximately 113.04 square centimeters. This is the calculated area of the compact disc. Now, let's examine the provided options and identify the one that matches our calculated area. The options are:
A. 36 square centimeters B. 452.16 square centimeters C. 113.04 square centimeters
Comparing our calculated area of approximately 113.04 square centimeters with the options, we can clearly see that option C, 113.04 square centimeters, is the correct answer. Options A and B are significantly different from our calculated area and can be easily ruled out. Option A is simply the square of the radius and doesn't account for π, while option B seems to be an incorrect calculation altogether. Therefore, the correct answer is C. 113.04 square centimeters. This concludes our calculation and selection process. We have successfully applied the formula for the area of a circle, performed the necessary calculations, and identified the correct answer from the given options. However, the process doesn't end here. It's important to reflect on the steps we took and ensure that our answer makes sense in the context of the problem. This involves checking our calculations, ensuring that the units are correct, and considering whether the answer is reasonable given the size of the compact disc. In the next section, we'll discuss the importance of verifying our answer and how to do so effectively.
Verifying the Answer
After calculating the area and selecting the answer, it's crucial to verify the result. Verifying the answer ensures accuracy and builds confidence in the solution. There are several ways to verify the answer in this case. One method is to use a more precise value of π (pi) and recalculate the area. We initially used 3.14 as an approximation for π. If we use a more precise value, such as 3.14159, we can recalculate the area to see if it aligns with our previous result. This will help us ensure that our approximation didn't significantly affect the final answer. Another method is to estimate the area. We know the radius is 6 centimeters. We can visualize a square with sides equal to the diameter of the circle (12 centimeters). The area of this square would be 12 cm * 12 cm = 144 square centimeters. Since the circle fits inside this square, its area must be less than 144 square centimeters. Our calculated area of 113.04 square centimeters is indeed less than 144 square centimeters, which provides a basic check on the reasonableness of our answer. Additionally, we can compare our answer with known values. For instance, we might have a general sense of the size of a compact disc and its approximate area. Our calculated area should align with this general understanding. If our calculated area was significantly larger or smaller than what we would expect for a compact disc, it would indicate a potential error in our calculations. Finally, we can double-check our calculations step by step. This involves revisiting each step of the calculation process, from substituting the radius into the formula to performing the multiplication, to ensure that we didn't make any arithmetic errors. By employing these verification methods, we can confidently confirm that our answer of 113.04 square centimeters is correct. In the next section, we'll discuss the real-world applications of calculating the area of a circle and how this skill can be useful in various contexts.
Real-World Applications
Understanding how to calculate the area of a circle has numerous real-world applications. This fundamental concept is used in various fields, from engineering and architecture to everyday situations. In engineering, calculating the area of circles is crucial in designing circular components, such as pipes, gears, and wheels. For example, engineers need to determine the cross-sectional area of pipes to calculate the flow rate of fluids or the load-bearing capacity of circular structures. In architecture, circular areas are important in designing circular buildings, domes, and other architectural features. Architects use these calculations to determine the amount of material needed for construction, the space available within a circular structure, and the aesthetic proportions of circular elements. In everyday life, we encounter circular areas in various contexts. For instance, when planning a garden, we might need to calculate the area of a circular flower bed to determine how many plants we can fit or how much fertilizer we need. When baking, we might need to calculate the area of a circular cake pan to determine the amount of batter required. Calculating the area of a circle is also essential in many scientific fields. For example, in astronomy, scientists use circular area calculations to determine the size of celestial objects, such as planets and stars. In physics, circular areas are used in calculations involving circular motion, such as the area swept by a rotating object. Furthermore, the concept of circular area extends to three-dimensional shapes, such as cylinders and spheres. The formulas for the volume and surface area of these shapes involve the area of a circle. Therefore, understanding how to calculate the area of a circle is a foundational skill for solving a wide range of mathematical and practical problems. The ability to accurately calculate circular areas is not only important for academic success but also for real-world problem-solving and decision-making. This skill empowers us to understand and interact with the world around us in a more informed and effective way. In the next section, we'll summarize the key concepts discussed in this article and highlight the importance of mastering this fundamental geometric concept.
Conclusion
In this article, we've explored the process of calculating the area of a circle, using the example of a compact disc. We've seen how the formula A = πr² is applied, and how the radius is the key dimension in determining the area. Mastering this concept is essential for various mathematical and real-world applications. We began by understanding the problem, identifying the given information (the radius), and recognizing that we needed to find the area of the circle. We then introduced the formula A = πr², which is the foundation for calculating the area of any circle. Next, we applied the formula, substituting the radius (6 centimeters) into the equation and performing the necessary calculations. We used the approximation π ≈ 3.14 to simplify the calculation, resulting in an area of approximately 113.04 square centimeters. After calculating the area, we compared our result with the given options and correctly identified option C (113.04 square centimeters) as the answer. We then emphasized the importance of verifying the answer. We discussed several methods for verification, including using a more precise value of π, estimating the area, and double-checking our calculations. These methods help ensure the accuracy of our solution and build confidence in our understanding. Finally, we explored the real-world applications of calculating the area of a circle. We saw how this skill is used in engineering, architecture, everyday life, and various scientific fields. From designing circular structures to calculating the size of celestial objects, the concept of circular area is universally applicable. In conclusion, understanding how to calculate the area of a circle is a fundamental skill that has wide-ranging applications. By mastering this concept, we can solve mathematical problems, make informed decisions in everyday life, and appreciate the geometry that surrounds us. This article has provided a comprehensive guide to calculating the area of a circle, from understanding the formula to verifying the answer and recognizing its real-world significance. We encourage you to continue practicing and applying this skill to further enhance your understanding and problem-solving abilities.
