A circle with a radius of 7 cm is inscribed in a square. What is the area of the square not covered by the circle?
["Title: Calculating the Area of Square Not Covered by an Inscribed Circle: A Step-by-Step Guide", "When geometry meets practical application, one fascinating problem arises: determining the area of a square that remains uncovered by a circle perfectly inscribed within it. In this article, we explore a classic example—a circle with a radius of 7 cm inscribed in a square—and reveal how to calculate the unused area in the square. Whether you're a student, educator, or geometry enthusiast, this explainer will illuminate key principles with clear steps and precise calculations.", "---", "### The Setup: Circle Inscribed in a Square", "Imagine a circle with a radius of 7 cm perfectly fitted inside a square. When a circle is inscribed in a square, the diameter of the circle equals the side length of the square. Since the radius is 7 cm, the diameter—and thus the side length of the square—is:", "[\n\ ext{Side length} = 2 \ imes \ ext{Radius} = 2 \ imes 7 = 14\ \ ext{cm}\n]", "---", "### Step 1: Compute the Area of the Square", "The area of a square is calculated using the formula:", "[\n\ ext{Area}{\ ext{square}} = \ ext{Side}^2 = 14^2 = 196\ \ ext{cm}^2\n]", "---", "### Step 2: Compute the Area of the Circle", "The area of a circle is given by:", "[\n\ ext{Area}^2}} = \pi \ imes r^2 = \pi \ imes 7^2 = \pi \ imes 49 \approx 153.94\ \ ext{cm\n]", "(Using (\pi \approx 3.1416) for precision.)", "---", "### Step 3: Calculate the Uncovered Area", "The area of the square not covered by the circle is simply the difference between the square’s area and the circle’s area:", "[\n\ ext{Uncovered Area} = \ ext{Area}{\ ext{square}} - \ ext{Area}^2}} = 196 - 153.94 = 42.06\ \ ext{cm\n]", "---", "### Visual Summary", "| Shape | Area (cm²) |\n|----------------|------------------|\n| Square (side 14 cm) | 196 |\n| Circle (r = 7 cm) | 49π ≈ 153.94 |\n| Uncovered Area | 196 – 49π ≈ 42.06 |", "---", "### Why This Matters", "This problem demonstrates a fundamental geometric relationship: the efficiency of inscribed shapes. The uncovered corner areas highlight how perfect fitting shapes still leave gaps. This concept applies in engineering, architecture, and design where material optimization is crucial.", "---", "### Final Answer", "The area of the square not covered by the inscribed circle is:", "[\n\boxed{196 - 49\pi\ \ ext{cm}^2 \approx 42.06\ \ ext{cm}^2}\n]", "Whether you’re calculating areas for homework or deepening spatial reasoning skills, mastering inscribed shapes builds a strong foundation in geometry—one with real-world relevance.", "---", "Keywords: inscribed circle, square area, circle uncovered area, geometry problem, radius 7 cm, inscribed circle and square, math problem solution, area calculation, π in geometry"]
