Solving for \( r \), \( r = \frac{C}{2\pi} = \frac{31.4}{2 \times 3.14159} \approx \frac{31.4}{6.28318} \approx 5 \).
["# Solving for ( r ): A Clear Breakdown of the Radius Calculation Using ( C )", "When working with circular geometry, understanding how to solve for radius ( r ) is essential. One commonly encountered formula relates the circumference ( C ) of a circle to its radius:", "[ r = \frac{C}{2\pi} ]", "This equation expresses that the radius is the circumference divided by ( 2\pi ), a fundamental relationship derived from the definition of ( \pi ), the ratio of a circle’s circumference to its diameter.", "## Why Solve for ( r )?", "In engineering, physics, and everyday applications, determining the radius from known circumference is crucial. Whether measuring a pipe’s internal diameter or calculating pond sizes, accurate radius computation ensures precision and avoids costly errors.", "## The Formula Explained", "Given:\n[ r = \frac{C}{2\pi} ]\nWith standard values, ( \pi \approx 3.14159 ), so:\n[ r = \frac{31.4}{2 \ imes 3.14159} ]", "Performing the full calculation:\n[ 2 \ imes 3.14159 = 6.28318 ]\n[ r = \frac{31.4}{6.28318} \approx 5 ]", "This confirms that a circumference of approximately 31.4 units yields a radius of about 5 units.", "## Step-by-Step Example", "Step 1: Recall the formula ( r = \frac{C}{2\pi} ).\nStep 2: Substitute known values: ( C = 31.4 ), ( \pi \approx 3.14159 ).\nStep 3: Compute the denominator: ( 2\pi = 2 \ imes 3.14159 = 6.28318 ).\nStep 4: Divide circumference by denominator: ( \frac{31.4}{6.28318} \approx 5 ).", "## Real-World Applications", "- Manufacturing: Calculating the radius of rolled metal sheets ensures materials fit design specifications.\n- Astronomy: Approximating planetary orbits based on measured circumferences gives approximate planetary radii.\n- Education: Teaching students to solve for ( r ) solidifies conceptual understanding of circular motion and geometry.", "## Conclusion", "Solving for radius ( r ) in terms of circumference is a straightforward yet vital calculation. Using ( r = \frac{C}{2\pi} ), plugging in values like ( C = 31.4 ) and ( \pi \approx 3.14159 ) leads quickly to ( r \approx 5 ). This simple solution underpins accurate measurements across science, engineering, and daily life — proving why foundational math remains indispensable.", "---", "Keywords: solve for ( r ), circular radius formula, circumference to radius, ( C = 2\pi r ), mathematical calculation, geometry definition, ( \pi \approx 3.14159 ), radius computation."]
