#### 4160Question: A regular hexagon is inscribed in a circle. If the side length of the hexagon is 5 cm, what is the circumference of the circle?
["# Understanding the Relationship Between a Regular Hexagon and Its Circumscribed Circle", "A regular hexagon inscribed in a circle is a classic geometric scenario that reveals a powerful mathematical relationship between polygons and circles. When the side length of a regular hexagon is known, so is the radius — and from that, the circumference of the circle. One common question explores: If the side length of a regular hexagon is 5 cm, what is the circumference of the circle in which it is inscribed? This article dives into the solution, the geometry behind it, and why this elegant relationship matters.", "## Why a Regular Hexagon Inscribed in a Circle is Special", "A regular hexagon has six equal sides and six equal angles. When inscribed in a circle, each of its vertices touches the circle, meaning all vertices lie exactly on the circumference. In this unique configuration, the side length of the hexagon directly equals the radius of the circumscribed circle. This key geometric property simplifies many calculations in geometry.", "## Step-by-Step: Finding the Circumference", "### Step 1: Recall the property of a regular hexagon\nIn a regular hexagon inscribed in a circle, the length of each side is equal to the radius of the circle.\nGiven:\nSide length of hexagon = 5 cm\nTherefore,\nRadius ( r = 5 ) cm", "### Step 2: Use the formula for circumference\nThe circumference ( C ) of a circle is given by:\n[\nC = 2\pi r\n]\nSubstitute ( r = 5 ) cm:\n[\nC = 2\pi \ imes 5 = 10\pi \ ext{ cm}\n]", "### Step 3: Express the value numerically (optional)\nUsing ( \pi \approx 3.1416 ),\n[\nC \approx 31.42 \ ext{ cm}\n]\nHowever, leaving it in terms of ( \pi ) maintains precision and is preferred in formal contexts.", "## Conclusion: The Circumference of the Circle is ( 10\pi ) cm", "For a regular hexagon inscribed in a circle with side length 5 cm, the circle’s circumference is exactly ( 10\pi ) cm. This simple relationship makes the regular hexagon one of the most instructive and aesthetically pleasing examples of symmetry between polygons and circles.", "Understanding this concept aids study in geometry, trigonometry, and even trigonometric applications involving periodicity and circular functions. Whether in academic settings or practical problems, recognizing this pattern saves time and highlights elegant mathematical truths.", "### Key Takeaways\n- Side length of inscribed regular hexagon = radius of circle\n- Formula for circumference: ( C = 2\pi r )\n- With side length 5 cm, circumference = ( 10\pi ) cm\n- This relationship is foundational in geometry and applied mathematics", "Explore more about polygons inscribed in circles — each shape reveals its own unique connection to the circle’s circumference!"]
