A circle is inscribed in a square with side length 10 cm. What is the area of the circle?
["Understanding Geometry: A Circle Inscribed in a Square with Side Length 10 cm", "When studying geometry, one of the fundamental shapes that often appears in problems is the relationship between a circle and a square—especially when a circle is inscribed in a square. A circle inscribed in a square touches all four sides of the square, perfectly fitting within its boundaries. This concept becomes particularly clear—and straightforward—when the square has equal sides, as in the case below.", "### What Does It Mean for a Circle to Be Inscribed in a Square?", "An inscribed circle in a square has a diameter equal to the length of the square’s side. This is because the circle touches each side of the square at its midpoint, so the diameter of the circle matches the side length of the square.", "In this example:", "- The side length of the square is 10 cm.\n- Therefore, the diameter of the inscribed circle is also 10 cm.\n- The radius is half the diameter: ( r = \frac{10}{2} = 5 ) cm.", "### How to Calculate the Area of the Inscribed Circle", "The area ( A ) of a circle is calculated using the formula:", "[\nA = \pi r^2\n]", "Substituting the radius:", "[\nA = \pi (5)^2 = 25\pi , \ ext{cm}^2\n]", "Using ( \pi \approx 3.14 ), the approximate area is:", "[\n25 \ imes 3.14 = 78.5 , \ ext{cm}^2\n]", "### Summary", "- Square side length: 10 cm\n- Diameter of inscribed circle: 10 cm\n- Radius of inscribed circle: 5 cm\n- Area of the circle: ( 25\pi , \ ext{cm}^2 ) (exact), approximately 78.5 cm² (decimal)", "The inscribed circle in a square with a 10 cm side provides a clear and elegant example of how geometric relationships allow us to calculate important properties like area using basic formulas. Whether for architecture, design, or pure math education, understanding this relationship deepens spatial reasoning and reinforces the beauty of geometry."]
