A circle with a diameter equal to the diagonal of a square with side length 5 cm is inscribed in a square. Find the area of the larger square.
["Title: Solving a Geometric Challenge: Circle Inscribed in a Larger Square Based on a Derived Square", "---", "Understanding geometric relationships can unlock elegant solutions in geometry. Today’s article explores a classic problem: determining the area of a larger square that circumscribes a circle inscribed using a square’s diagonal — all rooted in a square of known side length.", "---", "### The Setup: From a 5 cm Square to an Inscribed Circle", "We begin with a square of side length 5 cm. We’re told this square forms the basis for constructing a circle whose diameter equals the diagonal of the square.", "Step 1: Calculate the diagonal of the smaller square\nFor a square with side length ( s = 5 ) cm, the diagonal ( d ) is given by the Pythagorean theorem:", "[\nd = s\sqrt{2} = 5\sqrt{2} , \ ext{cm}\n]", "This diagonal becomes the diameter of the circle.", "---", "### Step 2: Circle Inscibed in the Larger Square", "The circle with diameter ( 5\sqrt{2} ) cm is inscribed in a larger square. When a circle is inscribed in a square, the diameter of the circle equals the side length of the square.", "Thus, the side length of the larger square is equal to the circle’s diameter:", "[\n\ ext{Side of larger square} = 5\sqrt{2} , \ ext{cm}\n]", "---", "### Step 3: Compute the Area of the Larger Square", "The area ( A ) of a square is the square of its side length:", "[\nA = (5\sqrt{2})^2 = 25 \ imes 2 = 50 , \ ext{cm}^2\n]", "---", "### Why This Geometric Insight Matters", "This problem beautifully combines foundational geometric principles:\n- Perfect symmetry of squares and circles\n- The role of diagonal in squares\n- Inscribed and circumscribed shapes", "By determining the circle’s diameter from a square’s diagonal and using it as the side length for the larger square, we efficiently found the area without complex coordinate geometry or advanced trigonometry.", "---", "### Final Answer", "The area of the larger square is 50 cm².", "---", "### Bonus Tip for Students\nAlways map geometric relationships carefully:\n- Identify known lengths\n- Use formulas step-by-step\n- Remember inscribed shapes: their diameter = side length of enclosing figure", "---", "Whether you’re solving Geometry problems in exams or understanding spatial relationships, mastering these foundational connections unlocks clarity and confidence.", "---", "Keywords: circle inscribed in square, diameter equals diagonal, square with side 5 cm, area of square, geometric problem solving, diagonal of square formula, square inscribed circle, geometry explanation", "Meta Description: Learn how a circle’s diameter equal to the diagonal of a 5 cm square leads to the area of a larger inscribing square — a step-by-step geometry solution with clear formulas and insights."]
