A circle is inscribed in a square with side length 10 cm. What is the area of the region outside the circle but inside the square?
["Incribed Circle in a Square: Calculating the Area Between the Circle and Square", "When a circle is inscribed in a square, it perfectly fits inside the square, touching the midpoint of each side. This classic geometric relationship offers a clear opportunity to explore area calculations and spatial reasoning. In this article, we’ll explore how a circle fits inside a square with a side length of 10 cm, and calculate the area of the region that lies outside the circle but still within the square.", "### Understanding the Geometry", "Imagine a square with each side measuring 10 cm. A circle inscribed in this square has a diameter equal to the side length of the square—so its diameter is 10 cm. Consequently, the circle’s radius is:", "$$\nr = \frac{10}{2} = 5 \ ext{ cm}\n$$", "### Area of the Square", "The area of a square is found using the formula:", "$$\n\ ext{Area}{\ ext{square}} = \ ext{side}^2 = 10^2 = 100 \ ext{ cm}^2\n$$", "### Area of the Inscribed Circle", "Using the circle’s radius, the area is calculated with the formula:", "$$\n\ ext{Area}^2}} = \pi r^2 = \pi \ imes 5^2 = 25\pi \ ext{ cm\n$$", "### Area Outside the Circle but Inside the Square", "To find the area of the region outside the circle but within the square, subtract the area of the circle from the area of the square:", "$$\n\ ext{Area}{\ ext{outside circle}} = \ ext{Area} = 100 - 25\pi}} - \ ext{Area}_{\ ext{circle}\n$$", "Using the approximate value $\pi \approx 3.1416$, we get:", "$$\n25\pi \approx 25 \ imes 3.1416 = 78.54 \ ext{ cm}^2\n$$", "$$\n100 - 78.54 = 21.46 \ ext{ cm}^2 \quad \ ext{(approximate)}\n$$", "Thus, the exact area is:", "$$\n100 - 25\pi \ ext{ cm}^2\n$$", "### Conclusion", "The region outside the inscribed circle but inside the square has an area of:", "$$\n\boxed{100 - 25\pi \ ext{ cm}^2 \approx 21.46 \ ext{ cm}^2}\n$$", "This elegant relationship between the square and its incircle is not only foundational in geometry but also widely used in fields like design, architecture, and engineering to calculate usable space within bounded shapes."]
