A circle is inscribed in a square with side length 14 cm. Calculate the area of the circle in square centimeters, then determine the area of the square not covered by the circle.
["Understanding Inscribed and Circumscribed Shapes: A Circle Inside a Square (14 cm Side Length)", "When a circle is inscribed in a square, it fits perfectly inside the square, touching all four sides. This geometric relationship offers a clean way to calculate areas and explore fundamental principles in mathematics. In this article, we’ll analyze a specific case: a circle inscribed in a square with a side length of 14 cm. We’ll calculate the area of the circle and determine the area remaining in the square not covered by the circle.", "---", "### What Does It Mean for a Circle to Be Inscribed in a Square?", "An inscribed circle in a square touches the midpoint of each side. The diameter of this circle equals the length of the square’s side. Since the square has a side length of 14 cm, the diameter of the inscribed circle is also 14 cm. Therefore, the radius of the circle is half that:\n[ \ ext{Radius} = \frac{14}{2} = 7 \ ext{ cm} ]", "---", "### Calculating the Area of the Inscribed Circle", "The area ( A ) of a circle is given by the formula:\n[ A = \pi r^2 ]\nSubstituting the radius ( r = 7 ) cm:\n[\nA = \pi \ imes 7^2 = \pi \ imes 49 = 49\pi \ ext{ cm}^2\n]", "Using the common approximation ( \pi \approx 3.14 ):\n[ A \approx 49 \ imes 3.14 = 153.86 \ ext{ cm}^2 ]", "So, the area of the inscribed circle is approximately 153.86 cm².", "---", "### Calculating the Area of the Square", "The square has side length 14 cm, so its area is:\n[ \ ext{Area}{\ ext{square}} = 14 \ imes 14 = 196 \ ext{ cm}^2 ]", "---", "### Calculating the Area Not Covered by the Circle", "To find the area of the square not covered by the circle, subtract the circle’s area from the square’s area:\n[\n\ ext{Uncovered Area} = \ ext{Area}^2}} - \ ext{Area}_{\ ext{circle}} = 196 - 49\pi \ ext{ cm\n]", "Using ( \pi \approx 3.14 ):\n[ \ ext{Uncovered Area} \approx 196 - 153.86 = 42.14 \ ext{ cm}^2 ]", "So, approximately 42.14 cm² of the square remains outside the inscribed circle.", "---", "### Summary", "- A circle inscribed in a square with side 14 cm has:\n - Radius = 7 cm\n - Area = ( 49\pi ) cm² (≈153.86 cm²)\n- The square has area = 196 cm²\n- The area of the square not covered by the circle:\n [ 196 - 49\pi \ ext{ cm}^2 \approx 42.14 \ ext{ cm}^2 ]", "This simple geometric relationship not only helps in visualizing shapes but also supports practical applications in design, calculations, and problem-solving across geometry and calculus.", "---", "### Key Takeaway", "When a circle is inscribed in a square:\n- The diameter of the circle equals the square’s side length.\n- The radius is half the side length.\n- The uncovered area reveals the elegant balance between curved and straight shapes in Euclidean geometry.", "Explore this classic configuration to deepen your understanding of circles, squares, and area relationships!"]
