Solution: The circumference $C = 2\pi r$, so $r = \frac{C}{2\pi} = \frac{12\pi}{2\pi} = 6$ nm.

["How to Calculate the Radius from Circumference: A Simple Guide", "Understanding fundamental geometric relationships is essential in science, engineering, and design. One of the most common applications is determining the radius of a circle when the circumference is known. Whether you're working in a lab, classroom, or technical field, knowing how to calculate the radius from circumference is straightforward—and incredibly useful.", "### The Formula: Circumference and Radius Relationship", "The circumference ( C ) of a circle is determined by the formula:", "[\nC = 2\pi r\n]", "where:\n- ( C ) = circumference\n- ( r ) = radius\n- ( \pi ) (pi) is a mathematical constant approximately equal to 3.14159", "Rearranging the formula to solve for the radius gives:", "[\nr = \frac{C}{2\pi}\n]", "This equation tells us that the radius is directly proportional to the circumference, scaled down by the factor ( 2\pi ).", "### Step-by-Step Calculation with an Example", "Suppose you measure the circumference to be ( C = 12\pi ) nanometers. To find the radius, apply the formula:", "[\nr = \frac{C}{2\pi} = \frac{12\pi}{2\pi} = 6 \ ext{ nm}\n]", "Here, the ( \pi ) terms cancel out, leaving a clean result: the radius is 6 nm. This illustrates how the relationship simplifies when circumference is expressed in terms of ( \pi ).", "### Why It Matters", "Calculating the radius from circumference is vital in many real-world contexts:", "- Engineering & Manufacturing: Determining wire diameter, pipe sizes, or circular components.\n- Physics & Astronomy: Finding orbital paths, wave wavelengths, or circular motion parameters.\n- Design & Architecture: Creating curved structures or ensuring precise material measurements.", "### Final Thoughts", "The formula ( r = \frac{C}{2\pi} ) is elegant and powerful—tying circle geometry together in a simple, accurate way. With just one step, you convert circumference into radius, unlocking valuable spatial understanding. Whether you’re solving a textbook problem or programming a scientific computation, remembering this relationship empowers precise, efficient work.", "---", "Key Takeaways:\n- Circumference ( C = 2\pi r )\n- Radius from circumference: ( r = \frac{C}{2\pi} )\n- Units matter: if ( C = 12\pi ) nm, then ( r = 6 ) nm\n- Useful in science, engineering, design, and more", "Master this simple equation, and you gain a fundamental tool for circular geometry!"]