Question: A circle is circumscribed around an equilateral triangle with side length \(6\) units. Find the radius of the circumscribed circle.
["# Finding the Radius of the Circumscribed Circle Around an Equilateral Triangle with Side Length 6 Units", "Understanding the relationship between an equilateral triangle and its circumscribed circle (circumcircle) is a fascinating topic in geometry. When a circle is circumscribed around an equilateral triangle, all three vertices of the triangle lie exactly on the circle’s boundary. The radius of this circle—called the circumradius—can be calculated using a well-known formula derived from geometric principles.", "If you’re given an equilateral triangle with side length (6) units, finding the radius (R) of the circumscribed circle is straightforward thanks to a simple mathematical relationship.", "## The Formula for the Circumradius of an Equilateral Triangle", "For any equilateral triangle with side length (a), the radius (R) of the circumscribed circle is given by:", "[\nR = \frac{a}{\sqrt{3}}\n]", "However, this form is sometimes expressed using a rationalized version for clarity. More accurately, the standard formula derived from triangle geometry and trigonometry is:", "[\nR = \frac{a}{\sqrt{3}} = \frac{a \sqrt{3}}{3}\n]", "Both forms are equivalent, and the second form—(\frac{a\sqrt{3}}{3})—is often preferred because it avoids division by (\sqrt{3}) and uses rational numbers.", "## Step-by-Step Calculation", "Given the side length (a = 6) units:", "1. Apply the circumradius formula:\n [\n R = \frac{6}{\sqrt{3}}\n ]", "2. Rationalize the denominator:\n [\n R = \frac{6}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{6\sqrt{3}}{3} = 2\sqrt{3}\n ]", "Thus, the radius of the circumcircle is (2\sqrt{3}) units.", "## Why This Formula Works", "In an equilateral triangle, all angles measure (60^\circ). The center of the circumcircle coincides with the centroid, circumcenter, and orthocenter due to the triangle’s symmetry. Using the law of sines, ( \frac{a}{\sin A} = 2R ), and substituting (a = 6) and (\angle A = 60^\circ):", "[\n2R = \frac{6}{\sin 60^\circ} = \frac{6}{\frac{\sqrt{3}}{2}} = \frac{12}{\sqrt{3}}\n]\n[\nR = \frac{6}{\sqrt{3}} = 2\sqrt{3}\n]", "This confirms the earlier result.", "## Summary", "- For an equilateral triangle of side (a = 6):\n- Circumradius (R = \frac{a\sqrt{3}}{3} = 2\sqrt{3}) units\n- This result is consistent across trigonometric and geometric proofs", "Whether designing architectural features, solving geometric puzzles, or studying triangle properties, knowing how to calculate the circumradius of an equilateral triangle empowers deeper insight.", "Key Takeaway: The radius of the circumscribed circle around an equilateral triangle with side length 6 units is ( \boxed{2\sqrt{3}} ) units."]
