The volume of a cone is 100π cubic centimeters, and its height is 12 cm. Find the radius of the base.
["# How to Find the Radius of a Cone Given Its Volume and Height — Step-by-Step Example", "When calculating geometric volume problems, one of the most common tasks is determining the base radius of a cone when the volume and height are known. In this article, we’ll demonstrate how to find the radius of a cone with a volume of 100π cm³ and a height of 12 cm, using the standard formula for the volume of a cone.", "## Keep reading to discover the full calculation and understand how this formula applies in real-world geometry and engineering applications.", "---", "## The Volume Formula for 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\n- $ r $ = radius of the base (in cm)\n- $ h $ = height of the cone (in cm)", "---", "## Step-by-Step Problem: Volume = $100\pi$ cm³, Height = 12 cm", "We are given:\n- Volume $ V = 100\pi \ ext{ cm}^3 $\n- Height $ h = 12 \ ext{ cm} $", "We need to find the radius $ r $.", "### Step 1: Substitute known values into the formula", "$$\n100\pi = \frac{1}{3} \pi r^2 (12)\n$$", "### Step 2: Simplify the right-hand side", "$$\n100\pi = \frac{12}{3} \pi r^2 = 4\pi r^2\n$$", "### Step 3: Divide both sides by $ \pi $", "$$\n100 = 4 r^2\n$$", "### Step 4: Solve for $ r^2 $", "$$\nr^2 = \frac{100}{4} = 25\n$$", "### Step 5: Take the square root", "$$\nr = \sqrt{25} = 5\n$$", "---", "## Final Answer", "The radius of the base of the cone is 5 cm.", "---", "## Why This Problem Matters", "Understanding how to calculate the radius of a cone using volume and height is essential in fields like architecture, civil engineering, and manufacturing, where precise measurements of tapered structures are required. The formula combines basic algebra with geometric principles, making it a fundamental skill in applied mathematics.", "For quick verification:\n- Volume: $ \frac{1}{3} \pi (5^2)(12) = \frac{1}{3} \pi (25)(12) = 100\pi $ cm³ — matches the given value.", "---", "### Key Takeaways", "- Use the cone volume formula: $ V = \frac{1}{3} \pi r^2 h $\n- Isolate $ r $ algebraically once known\n- Always verify by plugging results back into the original formula", "Whether for homework, a science project, or a professional project, mastering this calculation ensures accuracy and confidence in geometric problem-solving.", "---", "Keywords: cone volume formula, radius of a cone, geometry problem solution, calculate cone radius, pratical geometry, math tutorial, cone volume calculation, use V = 1/3 π r² h, 3 cm cone radius example."]
