The surface area \(A\) of a sphere is \(A = 4\pi r^2\). Solving for \(r\), \(314 = 4\pi r^2\) gives \(r^2 = \frac{314}{4\pi} \approx 25\), so \(r \approx 5\) centimeters.
["# Understanding the Surface Area of a Sphere: Solving for the Radius", "The surface area (A) of a sphere is given by the formula:\n[\nA = 4\pi r^2\n]\nThis fundamental equation is essential in geometry and has wide applications in physics, engineering, and everyday life—from calculating how much paint is needed to cover a basketball to designing advanced medical devices.", "## Why Solving for the Radius Matters", "While the surface area formula expresses area in terms of radius (r), many practical problems require you to reverse the process: given a specific surface area, determine the corresponding radius. Understanding how to solve for (r) strengthens your grasp of algebraic manipulation and mathematical modeling.", "## Step-by-Step Example: Finding the Radius from Surface Area", "Let’s explore a common problem where the surface area is given numerically, such as:\n[\n314 = 4\pi r^2\n]\nHere, (A = 314) square centimeters. We solve for (r) to find the sphere’s radius.", "### Step 1: Isolate (r^2)", "Start by dividing both sides of the equation by (4\pi):\n[\nr^2 = \frac{314}{4\pi}\n]\nUsing (\pi \approx 3.14), compute:\n[\nr^2 \approx \frac{314}{4 \ imes 3.14} = \frac{314}{12.56} \approx 25\n]", "### Step 2: Take the Square Root", "Now, take the square root of both sides to solve for (r):\n[\nr = \sqrt{25} = 5\n]\nSince radius must be positive, (r \approx 5) centimeters.", "### Verification", "Plugging (r = 5) back into the original formula:\n[\nA = 4\pi(5)^2 = 4\pi \ imes 25 = 100\pi \approx 314 \ ext{ cm}^2\n]\nThe result matches the given value, confirming the calculation.", "## Conclusion", "The formula (A = 4\pi r^2) provides a powerful tool for analyzing spherical surfaces. By skillfully solving for the radius—especially using approximate values—you unlock practical problem-solving abilities. Whether estimating material costs or understanding physical models, mastering this algebraic process enhances both mathematical proficiency and real-world application.", "If you're working with different values, simply rearrange the equation and use precise constants to compute radii accurately every time.", "---", "Key Terms: Surface area sphere, radius calculation, (A = 4\pi r^2), solving for (r), algebra example, geometry applications.\nKeywords for SEO: Surface area of sphere formula, how to solve for radius in sphere formula, r from surface area, sphere radius calculation, formulae and practice."]
