A cone has a volume of 100 cm³ and a height of 12 cm. Find the radius of the base.
["# Finding the Radius of a Cone’s Base When Volume and Height Are Known", "Understanding the geometry of a cone is essential in many fields, from engineering to everyday problem-solving. One frequently encountered question is how to determine the radius of a cone’s base when given its volume and height. In this article, we’ll explore how to solve for the radius using the cone volume formula, including a practical example where the volume is 100 cm³ and the height is 12 cm.", "## The Formula for the Volume of a Cone", "The volume ( V ) of a cone is calculated using the formula:", "[\nV = \frac{1}{3} \pi r^2 h\n]", "where:\n- ( V ) = volume (in cm³)\n- ( r ) = radius of the base (in cm)\n- ( h ) = height (in cm)", "This formula reflects the relationship between the base area (( \pi r^2 )) and the height, scaled by ( \frac{1}{3} ) due to the cone’s tapered shape.", "## Step-by-Step Solution for Given Values", "We are given:\n- Volume ( V = 100 ) cm³\n- Height ( h = 12 ) cm", "Substitute these values into the volume formula:", "[\n100 = \frac{1}{3} \pi r^2 (12)\n]", "Simplify the right-hand side:", "[\n100 = 4\pi r^2\n]", "Now solve for ( r^2 ):", "[\nr^2 = \frac{100}{4\pi} = \frac{25}{\pi}\n]", "Take the square root of both sides to find ( r ):", "[\nr = \sqrt{\frac{25}{\pi}} = \frac{5}{\sqrt{\pi}}\n]", "To express this in a decimal format (optional but helpful), approximate ( \pi \approx 3.1416 ):", "[\nr \approx \frac{5}{\sqrt{3.1416}} \approx \frac{5}{1.7725} \approx 2.82 \ ext{ cm}\n]", "## Final Answer", "The exact radius of the cone’s base is:", "[\nr = \frac{5}{\sqrt{\pi}} \ ext{ cm}\n]", "Or approximately 2.82 cm when using ( \pi \approx 3.1416 ).", "## Why This Matters", "Knowing how to compute the radius from volume and height enables accurate design, construction, and analysis in various applications. Whether calculating material needs for a manufacturing project or solving a math problem, mastering this formula is a valuable skill.", "For future reference:\n- Volume formula: ( V = \frac{1}{3} \pi r^2 h )\n- Step outlining and unit consistency are key to arriving at the correct answer.", "If you found this guide helpful, explore other geometric shapes and real-world applications of volume formulas—your understanding of spatial relationships just grew!"]
