A circle is inscribed in a square with a side length of 10 cm. What is the area of the shaded region outside the circle but inside the square?
["Title: Inscribed Circle in a Square: How to Calculate the Area of the Shaded Region (10 cm Side Length)", "---", "When a circle is inscribed perfectly inside a square, the circle touches all four sides at their midpoints. With the square having a side length of 10 cm, this geometric relationship allows us to compute both the area of the square and the area of the inscribed circle—key steps to finding the so-called shaded region: the space inside the square but outside the circle.", "### Understanding the Geometry", "In a square of side length ( s = 10 ) cm, the diameter of the inscribed circle is equal to the side length of the square. Therefore:", "- Diameter of the circle = 10 cm\n- Radius ( r ) = Diameter ÷ 2 = ( 10 \div 2 = 5 ) cm", "Since the circle fits snugly inside the square, the shaded region is the difference between the square’s area and the circle’s area.", "---", "### Step-by-Step Calculation", "#### 1. Area of the Square", "$$\n\ ext{Area}{\ ext{square}} = s^2 = 10^2 = 100 , \ ext{cm}^2\n$$", "#### 2. Area of the Inscribed Circle", "$$\n\ ext{Area}^2}} = \pi r^2 = \pi \ imes 5^2 = 25\pi , \ ext{cm\n$$", "#### 3. Area of the Shaded Region", "The shaded region is the difference:", "$$\n\ ext{Area}{\ ext{shaded}} = \ ext{Area}}} - \ ext{Area{\ ext{circle}} = 100 - 25\pi\n$$", "Using ( \pi \approx 3.1416 ):", "$$\n\ ext{Area}^2}} \approx 100 - 25 \ imes 3.1416 = 100 - 78.54 = 21.46 , \ ext{cm\n$$", "---", "### Final Answer", "The area of the shaded region outside the circle but inside the square is:", "$$\n\boxed{100 - 25\pi , \ ext{cm}^2} \quad \ ext{(exact)} \quad \ ext{or approximately} \quad \boxed{21.46 , \ ext{cm}^2}\n$$", "---", "### Conclusion", "Knowing how to calculate the area between an inscribed circle and the surrounding square not only sharpens geometry skills but also helps in solving real-world problems involving space optimization, design, and architectural planning. With a side length of 10 cm, the shaded area clearly demonstrates the harmony between precise measurements and elegant mathematical relationships."]
