Calculating Area And Circumference Sum For A Circle With Diameter 6

In this comprehensive guide, we'll delve into the fundamental properties of circles, specifically focusing on calculating the area and circumference given the diameter. We will explore a classic problem where we are given a circle with a diameter of 6 units and are tasked with finding the sum of its area (b square units) and circumference (c units). This problem serves as an excellent exercise in applying the core formulas related to circles and combining the results to arrive at the final answer. We'll meticulously walk through each step, providing clear explanations and insights to enhance your understanding of circle geometry. Let's embark on this journey of mathematical discovery and master the concepts of area and circumference calculations. We aim to provide a detailed and insightful solution that will not only help you solve this particular problem but also equip you with the knowledge to tackle similar challenges with confidence.

Consider a circle with a diameter of 6 units. This circle has an area denoted as b square units and a circumference denoted as c units. Our primary objective is to determine the value of the expression b + c. This problem encapsulates the core concepts of circle geometry and requires us to apply the relevant formulas effectively. The challenge lies in correctly calculating both the area and the circumference and then summing them to obtain the final answer. To successfully solve this problem, we must first understand the relationship between the diameter, radius, area, and circumference of a circle. We will then apply this knowledge to compute the values of b and c and, finally, find their sum. This exercise not only tests our understanding of formulas but also our ability to apply them in a practical context. Let’s delve into the solution step by step to unravel the answer. This problem will enhance your problem-solving skills and reinforce your understanding of key geometric principles.

To solve this problem, we need to recall the formulas for the area and circumference of a circle. The area (b) of a circle is given by the formula b = πr², where r is the radius of the circle. The circumference (c) of a circle is given by the formula c = 2πr, where r is the radius of the circle. We are given that the diameter of the circle is 6 units. The radius is half of the diameter, so the radius r is 6 / 2 = 3 units. Now we can calculate the area b and the circumference c. The area b is π*(3)² = 9π square units. The circumference c is 2π*(3) = 6π units. We are asked to find the value of b + c. So, b + c = 9π + 6π = 15π. Therefore, the value of b + c is 15π. This solution demonstrates the application of fundamental circle formulas and the importance of understanding the relationship between diameter, radius, area, and circumference. By breaking down the problem into smaller steps and applying the appropriate formulas, we can arrive at the correct answer. This process reinforces the basic principles of geometry and enhances our ability to solve mathematical problems efficiently.

Let's break down the calculation into detailed steps to ensure clarity and understanding. This step-by-step approach will help solidify the concepts and make the solution process more transparent. First, we identify the given information: the diameter of the circle is 6 units. From this, we can find the radius, which is half the diameter. So, the radius (r) is 6 / 2 = 3 units. Next, we recall the formula for the area of a circle: b = πr². Substituting the value of the radius, we get b = π*(3)² = π9 = 9π square units. Now, we recall the formula for the circumference of a circle: c = 2πr*. Substituting the value of the radius, we get c = 2π*(3) = 6π units. Finally, we add the area and the circumference: b + c = 9π + 6π = 15π. Thus, the value of b + c is 15π. This detailed step-by-step calculation highlights the methodical approach required to solve geometric problems. By breaking the problem into smaller, manageable parts, we can easily apply the relevant formulas and arrive at the correct solution. This meticulous process also helps in avoiding errors and ensures a clear understanding of the solution. Each step is crucial in building a strong foundation in geometry and problem-solving.

The value of b + c is 15π. Therefore, the correct answer is (B) 15π. This result is obtained by first calculating the area and circumference of the circle using the given diameter and then summing these two values. The process involves understanding the formulas for the area and circumference of a circle and applying them correctly. The final answer, 15π, represents the sum of the area (9π) and the circumference (6π) of the circle. This problem demonstrates the importance of precision and accuracy in mathematical calculations, as well as the need to understand the fundamental relationships between different geometric properties of a circle. The ability to solve such problems is crucial for building a strong foundation in mathematics and related fields. Mastering these concepts allows for the confident application of geometric principles in various practical scenarios. Accurate calculations and a clear understanding of formulas are key to success in such problems.

In conclusion, we have successfully determined that the value of b + c, where b is the area and c is the circumference of a circle with a diameter of 6 units, is 15π. This was achieved by first finding the radius, then calculating the area and circumference using the respective formulas, and finally summing the results. This problem serves as a valuable exercise in understanding and applying the fundamental properties of circles. The step-by-step approach used in the solution highlights the importance of breaking down complex problems into smaller, manageable parts. By mastering these basic concepts and techniques, we can confidently tackle more challenging problems in geometry and related fields. The ability to apply formulas correctly and perform accurate calculations is essential for success in mathematics. Moreover, this exercise reinforces the relationship between different geometric properties of a circle, such as the diameter, radius, area, and circumference. Continuous practice and a thorough understanding of the underlying principles are key to mastering these concepts and achieving proficiency in problem-solving.