In an equilateral triangle, the inradius \( r = \frac{s\sqrt{3}}{6} \), where \( s = 12 \).
["Understanding the Inradius of an Equilateral Triangle: Formula and Calculation with ( s = 12 )", "The inradius ( r ) of an equilateral triangle is a key geometric measurement that represents the radius of the inscribed circle tangent to all three sides. Accurately determining ( r ) helps in solving problems related to area, symmetry, and spatial efficiency in both mathematical theory and real-world applications. In an equilateral triangle, the elegant formula for the inradius follows from fundamental triangle properties and yields simplified results when side length ( s ) is known.", "### What is the Inradius?", "The inradius is the distance from the center of the inscribed circle (the incenter) to any side of the triangle. In an equilateral triangle—where all sides and angles are equal—the incenter coincides with the centroid, circumcenter, and orthocenter, making it a focal point of symmetry.", "### The Formula: ( r = \frac{s\sqrt{3}}{6} )", "For any equilateral triangle with side length ( s ), the inradius is given by:", "[\nr = \frac{s\sqrt{3}}{6}\n]", "This formula derives from the relationship between the triangle’s area, semiperimeter, and inradius. Since the area ( A ) of an equilateral triangle is ( \frac{\sqrt{3}}{4}s^2 ) and the semiperimeter is ( \frac{3s}{2} ), applying the standard inradius formula ( r = \frac{A}{s_{\ ext{semi}}} ) ultimately simplifies to the above expression.", "### Applying the Formula with ( s = 12 )", "Let’s substitute ( s = 12 ) into the formula:", "[\nr = \frac{12\sqrt{3}}{6} = 2\sqrt{3}\n]", "Thus, when each side of the equilateral triangle measures 12 units, the inradius is ( 2\sqrt{3} ), approximately equal to 3.464.", "### Significance and Applications", "Knowing the inradius helps in:", "- Calculating the area using ( A = r \cdot s_{\ ext{semi}} )\n- Designing structures or patterns involving equilateral triangles\n- Understanding geometric packing and tiling efficiency\n- Teaching foundational geometry concepts in education", "### Conclusion", "The formula ( r = \frac{s\sqrt{3}}{6} ) provides a concise and powerful way to determine the inradius of an equilateral triangle. With ( s = 12 ), the inradius simplifies neatly to ( 2\sqrt{3} ), reinforcing how symmetry simplifies mathematical computation. Whether solving geometry problems or exploring real-world shapes, this relationship is essential for accurate geometric analysis."]
