Lösen Sie nach r²: r² = 120π / (10π) = 12; r = √12 ≈ 3,46 cm.
["How to Solve r² = 120π / (10π): A Step-by-Step Guide to Find r and Understand the Result", "When solving equations involving ( r^2 ), simplification often leads to efficient and accurate results—no complicated formulas required. One clear example is solving the equation:\n[\nr^2 = \frac{120\pi}{10\pi} = 12\n]\nFrom here, finding ( r ) is straightforward and valuable for many real-world applications, especially in geometry, physics, and engineering.", "### Understanding the Simplification", "Start with the original equation:\n[\nr^2 = \frac{120\pi}{10\pi}\n]\nNotice that both \pi terms appear in the numerator and denominator—they cancel out cleanly:\n[\nr^2 = \frac{120\pi}{10\pi} = \frac{120}{10} = 12\n]\nThis cancellation makes the equation simple and reduces error in manual calculations.", "### Solving for ( r )", "Now that ( r^2 = 12 ), take the square root of both sides:\n[\nr = \sqrt{12}\n]\nTo simplify ( \sqrt{12} ), factor 12 into perfect squares:\n[\n\sqrt{12} = \sqrt{4 \ imes 3} = \sqrt{4} \ imes \sqrt{3} = 2\sqrt{3}\n]\nSince ( r ) represents a physical length (such as radius), we typically take the positive root. Thus:\n[\nr = 2\sqrt{3} \approx 3.46~\ ext{cm}\n]", "### Why This Calculation Matters", "Knowing how to solve for ( r ) from such equations is essential when calculating radii, areas, volumes, or other geometric properties. For example, if a circular component has a surface expression simplified to ( r^2 = 120\pi / (10\pi) ), correctly identifying ( r \approx 3.46~\ ext{cm} ) ensures precision in manufacturing or design.", "### Final Result", "[\nr^2 = \frac{120\pi}{10\pi} = 12 \quad \Rightarrow \quad r = \sqrt{12} = 2\sqrt{3} \approx 3.46~\ ext{cm}\n]", "Mastering these basic algebraic steps ensures clarity, speed, and accuracy in mathematical problem-solving and real-world applications.", "---", "Keywords: solve ( r^2 ), how to simplify ( \frac{120\pi}{10\pi} ), calculate ( r ), geometric radius calculation, algebra steps, ( r \approx 3.46 , \ ext{cm} ), simplifying square roots, ( 2\sqrt{3} ), area calculations, geometry tips.", "---", "Note: Avoiding complex symbols and using clear explanations makes mathematical reasoning accessible and practical—key principles in effective technical communication."]
