A sphere has a surface area of \( 144\pi \) square units. What is the radius of the sphere? Use the formula \( A = 4\pi r^2 \).
["# Finding the Radius of a Sphere with a Given Surface Area", "If you've ever wondered how to calculate the radius of a sphere when you know its surface area, you're in the right place. In this article, we’ll explore the mathematical relationship between a sphere’s surface area and its radius, using the formula:\n[ A = 4\pi r^2 ]\nwhere ( A ) is the surface area and ( r ) is the radius. Today, we’ll solve the specific problem: If a sphere has a surface area of ( 144\pi ) square units, what is its radius?", "## Understanding the Formula", "The formula for the surface area of a sphere is straightforward:\n[ A = 4\pi r^2 ]\nThis equation expresses the total area covering the entire outer surface of a sphere in terms of its radius. Since we are given ( A = 144\pi ), we can substitute this into the formula and solve for ( r ).", "## Step-by-Step Problem Solving", "### Step 1: Substitute the given surface area into the formula\nWe start with:\n[ 144\pi = 4\pi r^2 ]", "### Step 2: Simplify the equation\nDivide both sides by ( \pi ) to eliminate the constant:\n[ 144 = 4r^2 ]", "Then, divide both sides by 4:\n[ 36 = r^2 ]", "### Step 3: Solve for ( r )\nTake the square root of both sides to isolate ( r ):\n[ r = \sqrt{36} ]\n[ r = 6 ]", "Since radius cannot be negative, we discard the negative root.", "The radius of the sphere is 6 units.", "## Why This Formula Matters", "Understanding the surface area formula is essential in fields such as physics, engineering, and computer graphics, where precise geometric modeling is required. Whether designing a spherical tank, studying planetary shapes, or rendering 3D objects, knowing how radius relates to surface area enables accurate calculations and projections.", "### Conclusion", "To summarize, a sphere with a surface area of ( 144\pi ) square units has a radius of 6 units, derived cleanly from the formula ( A = 4\pi r^2 ). Mastering this kind of algebraic manipulation strengthens foundational geometry skills and supports more advanced applications.", "If you’re working with spheres, check your work by plugging the radius back into the formula:\n[ A = 4\pi (6)^2 = 4\pi \cdot 36 = 144\pi ]\nThis matches the given surface area, confirming our solution is correct.", "---\nKeywords: sphere surface area, how to find radius from surface area, formula A = 4πr², solve for r in sphere, geometry problems, 3D shape calculations"]
