Solution: The area of a circle is given by $ A = \pi r^2 $. Setting $ \pi r^2 = 25\pi $, divide both sides by $ \pi $ to get $ r^2 = 25 $. Taking the square root yields $ r = 5 $. \boxed{5}

["Understanding the Area of a Circle: How to Solve $ A = \pi r^2 $ When $ A = 25\pi $", "The area ( A ) of a circle is a fundamental concept in geometry, calculated using the formula:\n[ A = \pi r^2 ]\nThis equation states that the area is proportional to the square of the radius ( r ), with ( \pi ) as a constant linking linear and area measurements. But what if you’re given a specific area value—say, ( A = 25\pi )—and need to find the radius? Let’s break down the solution step-by-step.", "---", "### Step 1: Start with the Area Formula\nBegin with the general area formula:\n[\nA = \pi r^2\n]\nSubstitute the given area ( 25\pi ) into the equation:\n[\n25\pi = \pi r^2\n]", "---", "### Step 2: Divide Both Sides by ( \pi )\nTo isolate ( r^2 ), divide every term by ( \pi ) (since ( \pi <br/>\neq 0 )):\n[\n\frac{25\pi}{\pi} = \frac{\pi r^2}{\pi}\n]\nThis simplifies to:\n[\n25 = r^2\n]", "---", "### Step 3: Solve for ( r ) by Taking the Square Root\nNow that you have ( r^2 = 25 ), apply the square root to both sides:\n[\nr = \sqrt{25}\n]\nSince radius represents a length, we only consider the positive root:\n[\nr = 5\n]", "---", "### Final Answer: The Radius Is 5\n[\n\boxed{5}\n]", "This means a circle with an area of ( 25\pi ) square units has a radius of exactly 5 units. This elegant solution highlights how algebraic manipulation—especially canceling common factors and applying square roots—simplifies geometric problems. Mastering this technique helps in both academic settings and real-world applications, from calculating material needs to designing circular structures.", "---", "Why This Equation Matters\nUnderstanding how to derive the radius from the area formula is essential for geometry, architecture, physics, and engineering. It reinforces the deep relationship between radius and area, empowering learners to tackle related problems confidently.", "Whether you're solving for ( r ) in class, designing a circular garden, or coding geometry-based simulations, remembering that ( r = \sqrt{A / \pi} ) opens the door to accurate and efficient calculations every time.\nThe radius of the circle is ( \boxed{5} )."]