A circle is circumscribed around a square with side length \(5 \, \text{cm}\). What is the circumference of the circle?

["# Finding the Circumscribed Circle’s Circumference Around a Square with Side Length 5 cm", "When solving geometry problems involving squares and circles, one important concept is the circumcircle—a circle that passes through all four vertices of a polygon. In this article, we explore what happens when a circle is circumscribed around a square with side length (5 , \ ext{cm}), and specifically determine the circumference of that circle.", "## What does it mean for a circle to be circumscribed around a square?", "A circumscribed circle around a square touches all four vertices of the square. This means the circle is centered at the square’s center, and the diagonal of the square is equal to the diameter of the circle.", "## Step 1: Find the diagonal of the square", "Let the side length of the square be ( s = 5 , \ ext{cm} ).\nThe diagonal ( d ) of a square can be calculated using the Pythagorean theorem:\n[\nd = s\sqrt{2}\n]\nSubstituting ( s = 5 ):\n[\nd = 5\sqrt{2} , \ ext{cm}\n]", "## Step 2: Determine the radius of the circumscribed circle", "Since the diagonal of the square equals the diameter of the circumcircle:\n[\n\ ext{Diameter} = 5\sqrt{2} , \ ext{cm}\n]\nThe radius ( r ) is half the diameter:\n[\nr = \frac{5\sqrt{2}}{2} , \ ext{cm}\n]", "## Step 3: Calculate the circumference of the circle", "The circumference ( C ) of a circle is given by the formula:\n[\nC = 2\pi r\n]\nSubstituting the radius:\n[\nC = 2\pi \left( \frac{5\sqrt{2}}{2} \right) = 5\sqrt{2}\pi , \ ext{cm}\n]", "## Final Answer", "The circumference of the circle circumscribed around a square with side length (5 , \ ext{cm}) is:\n[\n\boxed{5\sqrt{2}\pi , \ ext{cm}}\n]", "Understanding this relationship helps in solving various geometric layouts, from architecture to design, where squares and circumscribed circles often appear together."]