Finding The Radius Of A Circle With Circumference 12π A Step-by-Step Solution
In the realm of mathematics, circles hold a special significance, their elegant curves and symmetrical properties captivating mathematicians and enthusiasts alike. One of the fundamental properties of a circle is its circumference, the distance around the circle. The relationship between a circle's circumference (C) and its radius (r) is elegantly expressed by the formula C = 2πr, where π (pi) is a mathematical constant approximately equal to 3.14159. This formula serves as a cornerstone for solving various problems related to circles, including determining the radius when the circumference is known.
Step 1: Laying the Foundation: The Circumference Formula
Our journey begins with the fundamental formula that governs the relationship between a circle's circumference and its radius: C = 2πr. This equation is the bedrock upon which we'll build our solution. The formula states that the circumference (C) of a circle is directly proportional to its radius (r), with the constant of proportionality being 2π. This means that if we know the circumference of a circle, we can use this formula to determine its radius, and vice versa.
Understanding this formula is crucial for navigating the world of circles and their properties. It allows us to connect the linear distance around the circle (circumference) to the radial distance from the center to the edge (radius). This connection is essential for solving a wide range of problems, from calculating the amount of fencing needed for a circular garden to determining the size of a circular gear in a machine.
The formula C = 2πr is not just a mathematical equation; it's a gateway to understanding the geometry of circles. It allows us to quantify the relationship between a circle's size and its dimensions, paving the way for further exploration of circular shapes and their applications in various fields.
Step 2: Plugging in the Given Value: Substituting the Circumference
With the foundation laid, we move to the next step: substituting the given circumference value into the formula. We are given that the circumference of the circle is 12π. By substituting this value into the formula C = 2πr, we get: 12π = 2πr. This equation now contains only one unknown, the radius (r), which we aim to determine.
This step is crucial because it bridges the gap between the abstract formula and the specific problem we are trying to solve. By plugging in the given value, we transform the general equation into a concrete one, ready for algebraic manipulation.
Substitution is a fundamental technique in mathematics, allowing us to replace variables with their corresponding values. This process is essential for solving equations and applying mathematical principles to real-world scenarios. In this case, substituting the circumference value allows us to isolate the radius and solve for its value.
Step 3: Isolating the Radius: Dividing Both Sides by 2π
To isolate the radius (r), we need to eliminate the coefficient 2π from the right side of the equation. This can be achieved by dividing both sides of the equation by 2π. This operation maintains the equality of the equation while bringing us closer to solving for r. Performing this division, we get: (12π) / (2π) = (2πr) / (2π). This step simplifies the equation and isolates the term containing the radius.
The principle behind this step is the fundamental algebraic rule that states that we can perform the same operation on both sides of an equation without changing its balance. This rule is the cornerstone of equation solving, allowing us to manipulate equations strategically to isolate the variable we are interested in.
Dividing both sides by 2π is a key step in solving for the radius because it cancels out the 2π term on the right side, leaving us with the radius multiplied by 1. This brings us one step closer to finding the value of r.
Step 4: Simplifying the Equation: Performing the Division
Now, let's simplify the equation by performing the division on both sides. On the left side, (12π) / (2π) simplifies to 6. On the right side, (2πr) / (2π) simplifies to r. Thus, the equation becomes: 6 = r. This step reveals the numerical relationship between the simplified terms, but it seems we've encountered an error. The equation 6 = πr is the correct simplification after step 3, not 6 = r.
It's important to carefully review each step to ensure accuracy. In this case, the simplification in step 4 seems to have skipped a crucial division. We need to continue isolating 'r' by dividing by π.
Mathematical problem-solving often involves a series of steps, and it's crucial to verify each step to avoid errors. Even a small mistake can lead to an incorrect final answer. Therefore, it's always a good practice to double-check your work and ensure that each step is logically sound.
Step 5: Correcting the Course: Dividing by Pi to Isolate r
Recognizing the error in the previous step, we need to take corrective action. The equation we arrived at was 6 = πr. To isolate 'r', we must divide both sides of the equation by π. This gives us: 6/π = (πr)/π. This step correctly isolates the radius on one side of the equation.
This correction highlights the importance of careful attention to detail in mathematical problem-solving. Even experienced mathematicians can make mistakes, and the key is to identify and correct them promptly. This process often involves reviewing the steps taken and ensuring that each operation is mathematically sound.
The act of correcting an error is a valuable learning experience. It reinforces the importance of accuracy and provides an opportunity to deepen our understanding of the underlying concepts. In this case, the correction reinforces the principle of isolating a variable by performing inverse operations on both sides of the equation.
Step 6: Finding the Solution: The Radius of the Circle
Finally, performing the division, we find the value of the radius: r = 6/π. This is the exact value of the radius. To obtain an approximate decimal value, we can use the approximation π ≈ 3.14159. Dividing 6 by this value gives us an approximate radius of 1.90986. Therefore, the radius of the circle with a circumference of 12π is approximately 1.90986 units.
This final step represents the culmination of our problem-solving journey. We have successfully applied the formula for the circumference of a circle, performed algebraic manipulations, and arrived at the solution. The radius, expressed as 6/π, or approximately 1.90986 units, is the answer we sought.
Finding the solution is not just about arriving at a numerical answer; it's about the process of understanding the problem, applying the relevant concepts, and systematically working towards a resolution. This process is a valuable skill that extends far beyond the realm of mathematics.
Discussion and Conclusion
In this step-by-step solution, we've demonstrated how to determine the radius of a circle when its circumference is known. The process involves understanding the relationship between circumference and radius, expressed by the formula C = 2πr, and applying algebraic techniques to isolate the desired variable. The key steps include substituting the given circumference value, dividing both sides of the equation by 2π, and then dividing by π to finally isolate the radius. The radius of the circle with a circumference of 12π is 6/π, which is approximately 1.90986 units.
This problem exemplifies the power of mathematical formulas and the importance of algebraic manipulation in solving geometric problems. By understanding the relationships between different geometric properties and applying the appropriate formulas, we can unlock a wealth of knowledge about shapes and their characteristics. The process of solving this problem also highlights the importance of careful attention to detail and the ability to identify and correct errors along the way. Mathematical problem-solving is not just about finding the right answer; it's about developing critical thinking skills and a systematic approach to problem-solving that can be applied to various aspects of life.
