A circle is inscribed in a square with side length 10 cm. What is the area of the shaded region outside the circle but inside the square?
["Title: Area of the Shaded Region: Circle Inscribed in a Square – A Detailed Calculation", "---", "When a circle is perfectly inscribed inside a square, every side of the square touches the circle at exactly one point—meaning the circle fits snugly within the square, touching all four sides. In this problem, we’re given a square with a side length of 10 cm, and a circle perfectly inscribed within it. This creates a visually pleasing geometric relationship—and an opportunity to explore the area between the square and the circle, also known as the shaded region.", "### How to Find the Area of the Shaded Region", "The area of the shaded region is the difference between the area of the square and the area of the inscribed circle.", "#### Step 1: Calculate the area of the square\nThe square has a side length of 10 cm:\n[\n\ ext{Area}{\ ext{square}} = \ ext{side}^2 = 10^2 = 100~\ ext{cm}^2\n]", "#### Step 2: Determine the radius and area of the inscribed circle\nSince the circle is inscribed in the square, its diameter equals the side length of the square. Therefore:\n[\n\ ext{Diameter} = 10~\ ext{cm} \Rightarrow \ ext{Radius} = \frac{10}{2} = 5~\ ext{cm}\n]\nUsing the formula for the area of a circle, (A = \pi r^2):\n[\n\ ext{Area}^2}} = \pi \ imes 5^2 = 25\pi~\ ext{cm\n]", "#### Step 3: Compute the area of the shaded region\nThe shaded area is the part of the square not covered by the circle:\n[\n\ ext{Shaded Area} = \ ext{Area}{\ ext{square}} - \ ext{Area}^2}} = 100 - 25\pi~\ ext{cm\n]", "---", "### Final Answer", "The area of the shaded region outside the circle but inside the square is:\n[\n\boxed{100 - 25\pi~\ ext{cm}^2}\n]\nFor a numerical approximation, since (\pi \approx 3.1416), this is roughly (100 - 78.54 = 21.46~\ ext{cm}^2).", "---", "### Why This Matters\nUnderstanding the relationship between inscribed shapes deepens geometric intuition and highlights practical applications in architecture, design, and engineering. It also reinforces key formulas for area, perimeter, and the famous ratio (\pi) in real-world contexts.", "---", "Optimizing this content with keywords such as “inscribed circle area,” “shaded region math,” and “square circle difference” helps attract readers searching for clear, accurate geometric problems and solutions—perfect for math students, educators, or anyone exploring geometry."]
