A square and a circle have the same perimeter. If the square’s side is 10 cm, what is the circle’s area?
["A Square and a Circle with the Same Perimeter: What’s the Circle’s Area if the Square’s Side is 10 cm?", "When it comes to geometry, comparing shapes with equal perimeters offers a fascinating way to explore mathematical relationships. In this article, we’ll dive into a classic problem: If a square and a circle have the same perimeter, what is the area of the circle when the square’s side is 10 cm?", "---", "### Understanding Perimeter Equality", "Let’s start by calculating the perimeter of the square. Since a square has four equal sides and each side is 10 cm:", "Perimeter of square = 4 × side = 4 × 10 = 40 cm", "For the circle to have the same perimeter, it must also have a circumference of 40 cm.", "The circumference ( C ) of a circle is given by:", "[\nC = 2\pi r\n]", "where ( r ) is the radius. Setting this equal to 40:", "[\n2\pi r = 40\n]", "Solving for ( r ):", "[\nr = \frac{40}{2\pi} = \frac{20}{\pi} \ ext{ cm}\n]", "---", "### Finding the Circle’s Area", "Now that we know the circle’s radius, we can calculate its area ( A ) using the formula:", "[\nA = \pi r^2\n]", "Substitute ( r = \frac{20}{\pi} ):", "[\nA = \pi \left(\frac{20}{\pi}\right)^2 = \pi \cdot \frac{400}{\pi^2} = \frac{400}{\pi}\n]", "Using ( \pi \approx 3.1416 ), the area is approximately:", "[\nA \approx \frac{400}{3.1416} \approx 127.32 \ ext{ cm}^2\n]", "But the exact value, without approximation, is:", "[\n\boxed{\frac{400}{\pi} \ ext{ cm}^2}\n]", "---", "### Key Takeaways", "- When a square and a circle share the same perimeter, their shapes balance perimeter and area differently—squares have fixed corners, while circles maximize area for a given perimeter.\n- Knowing the perimeter (40 cm) allows us to precisely calculate the circle’s radius and area.\n- The circle’s area being ( \frac{400}{\pi} ) cm² shows how a single value emerges from two seemingly different shapes sharing a common boundary.", "---", "### Why This Matters", "This problem illustrates fundamental geometric principles: perimeter defines a shape’s boundary, while area reflects the space enclosed. Comparing shapes via perimeter helps engineers, architects, and designers optimize space and material use.", "If you’re studying geometry, embracing such perimeter-and-area puzzles strengthens your understanding of mathematical relationships and real-world applications.", "---", "Keywords: square perimeter and circle, circumference and radius, area of circle formula, geometry problem with equal perimeter, square side 10 cm, circle area calculation, ( \frac{400}{\pi} ) cm², perimeter and area relationships", "---", "Need help solving more geometry problems? Explore step-by-step solutions and insights in our full geometry guide!"]
