\( 31.4 = 2 imes 3.14 imes r \Rightarrow r = rac{31.4}{6.28} = 5 \) cm.

["# Solving ( 31.4 = 2 \ imes 3.14 \ imes r ): A Simple Breakdown", "Understanding algebraic equations is easier with clear examples — and the equation ( 31.4 = 2 \ imes 3.14 \ imes r ) offers a straightforward illustration of finding an unknown variable ( r ) using basic arithmetic and algebra. In this article, we’ll explore how solving this equation leads naturally to ( r = 5 ) cm, making it ideal for students, educators, and anyone interested in foundational math.", "## What the Equation Means", "The equation\n[ 31.4 = 2 \ imes 3.14 \ imes r ]\ntells us that a total value (31.4) equals the product of a constant (2), a known multiplier (3.14), and an unknown length ( r ). In practical terms, this could model real-world scenarios such as calculating the radius of a circle when given the area, where ( 3.14 ) approximates ( \pi ).", "## Step-by-Step Solution", "### Step 1: Recognize the components\nWe rewrite the equation for clarity:\n[ 31.4 = (2 \ imes 3.14) \ imes r ]\nMultiply ( 2 \ imes 3.14 = 6.28 ), so:\n[ 31.4 = 6.28 \ imes r ]", "### Step 2: Isolate ( r )\nTo solve for ( r ), divide both sides by 6.28:\n[ r = \frac{31.4}{6.28} ]", "### Step 3: Perform the division\nUsing division:\n[ r = 5 ]", "So,\n[ r = 5 \ ext{ cm} ]", "## Why This Matters: The Geometry Connection", "The number ( 3.14 ) is widely recognized as an approximation of the mathematical constant ( \pi ), approximately equal to 3.14159. When multiplied by 2, it forms ( 6.28 ), which represents the circumference of a circle with radius 1. Therefore, the equation models:\n[ C = 2 \pi r = 6.28 \ imes r = 31.4 \ ext{ cm} ]\nSolving gives ( r = 5 ) cm — confirming the radius when the circumference is 31.4 cm.", "## Real-World Applications", "This method applies to numerous practical fields:", "- Engineering: Calculating pipe diameters or gear radii when circumference and material properties are known.\n- Manufacturing: Designing cylindrical components like tanks or shafts.\n- Education: Teaching algebraic manipulation through tangible geometric contexts.", "## Final Notes", "The equation\n[ 31.4 = 2 \ imes 3.14 \ imes r ]\ndemonstrates how multiplication and division work in tandem to isolate variables and uncover unknowns. With a simple calculation, we find the radius is exactly 5 cm — a strong example of math’s power in problem-solving.", "Whether you're a student mastering algebra or a professional using practical calculations, mastering this pattern builds confidence for more complex equations ahead.", "---", "Keywords:\n( r = 5 ) cm, ( 31.4 = 2 \ imes 3.14 \ imes r ), solving linear equations, algebra with geometry, circumference formula, solving for radius, practical math examples"]