A circle is inscribed in a square with a side length of 12 cm. Calculate the area of the shaded region outside the circle but inside the square.
["# The Geometry of a Square with an Inscribed Circle: Area of the Shaded Region Explained", "When a circle is inscribed perfectly inside a square, the circle touches all four sides of the square. This simple yet elegant geometric relationship forms the basis for many real-world applications and scholastic geometry problems. In this article, we explore a classic problem: when the side length of the square is 12 cm, what is the area of the shaded region that lies between the circle and the square?", "## Understanding the Relationship: Square and Its Inscribed Circle", "A square has four equal sides and four right angles. An inscribed circle fits perfectly inside the square, touching the midpoint of each side. The diameter of the inscribed circle is exactly equal to the side length of the square.", "Given:\nSide length of the square = 12 cm\nThus, Diameter of the circle = 12 cm\nRadius of the circle = Diameter ÷ 2 = 12 cm ÷ 2 = 6 cm", "## Step-by-Step Calculation of Areas", "### 1. Area of the Square\nThe formula for the area of a square is:\n[\n\ ext{Area}{\ ext{square}} = \ ext{side}^2\n]\n[\n\ ext{Area}^2}} = 12^2 = 144~\ ext{cm\n]", "### 2. Area of the Inscribed Circle\nThe area of a circle is calculated using the formula:\n[\n\ ext{Area}{\ ext{circle}} = \pi r^2\n]\nWith radius ( r = 6 ) cm:\n[\n\ ext{Area}^2}} = \pi \ imes 6^2 = \pi \ imes 36 = 36\pi~\ ext{cm\n]\nUsing ( \pi \approx 3.1416 ):\n[\n36\pi \approx 36 \ imes 3.1416 = 113.097~\ ext{cm}^2 \quad (\ ext{or } \approx 113.1~\ ext{cm}^2)\n]", "### 3. Area of the Shaded Region\nThe shaded region is the area inside the square but outside the circle. This is simply the difference between the area of the square and the area of the circle:", "[\n\ ext{Area}{\ ext{shaded}} = \ ext{Area}}} - \ ext{Area{\ ext{circle}} = 144 - 36\pi\n]", "In exact form:\n[\n\ ext{Area}^2}} = 144 - 36\pi~\ ext{cm\n]\nApproximately:\n[\n144 - 113.1 = 30.9~\ ext{cm}^2 \quad (\ ext{rounded to one decimal place})\n]", "## Summary", "When a circle is inscribed in a square with a side length of 12 cm:\n- The circle has a radius of 6 cm.\n- The square has an area of 144 cm².\n- The inscribed circle has an area of ( 36\pi ) cm², approximately 113.1 cm².\n- The shaded region, the area between the square and the circle, has an area of ( 144 - 36\pi ) cm², or about 30.9 cm².", "### Why This Matters\nThis problem illustrates fundamental geometric principles, helps reinforce understanding of area calculations, and introduces the concept of shaded regions useful in both academic learning and practical fields like architecture, engineering, and design.", "## Final Answer\nThe area of the shaded region outside the inscribed circle but inside the square is:\n[\n\boxed{144 - 36\pi~\ ext{cm}^2}\n]\n(approximately 30.9 cm²)"]
