Radius of inscribed circle (inradius) = (side × √3) / 6 = (10√3) / 6 = (5√3) / 3 cm

["# Inradius of an Equilateral Triangle: Formula, Derivation, and Practical Use (With Side = 10 cm)", "When studying geometry, one of the most elegant and fascinating concepts is the radius of the inscribed circle, commonly called the inradius. For equilateral triangles, this value simplifies beautifully due to their perfect symmetry. In this article, we’ll explore the formula for the inradius, why it equals ((\ ext{side} \ imes \sqrt{3}) / 6), and compute the inradius when the side length is (10,\ ext{cm})—resulting in (\frac{5\sqrt{3}}{3},\ ext{cm}).", "---", "## What Is the Inradius (Radius of the Inscribed Circle)?", "The inradius is the radius of the largest circle that can fit inside a polygon—specifically, one that is tangent to all its sides. In an equilateral triangle, where all sides and angles are equal, the inradius is both simple to compute and visually intuitive.", "---", "## The Formula: Radius = (Side × √3) / 6", "For any equilateral triangle with side length ( s ), the formula for the inradius ( r ) is:", "[\nr = \frac{s \sqrt{3}}{6}\n]", "### Why does this formula work?", "An equilateral triangle can be derived from a 30°–60°–90° triangle formed by dropping a perpendicular from one vertex to the midpoint of the opposite side—a height of the triangle. This height splits the equilateral triangle into two congruent right triangles.", "- Let the side length be ( s ).\n- The height ( h ) is:\n [\n h = \frac{s\sqrt{3}}{2}\n ]\n- The centroid (and incenter, since the triangle is equilateral) divides the height in a 2:1 ratio, placing the inradius at one-third of the height from the base:\n [\n r = \frac{1}{3} \cdot \frac{s\sqrt{3}}{2} = \frac{s\sqrt{3}}{6}\n ]", "This derivation shows how symmetry and trigonometry combine to yield a concise, elegant result.", "---", "## Applying the Formula: Case of Side = 10 cm", "Let’s apply the formula with ( s = 10,\ ext{cm} ):", "[\nr = \frac{10 \ imes \sqrt{3}}{6} = \frac{10\sqrt{3}}{6} = \frac{5\sqrt{3}}{3},\ ext{cm}\n]", "Thus, the radius of the inscribed circle in an equilateral triangle with side 10 cm is exactly:", "[\n\boxed{\frac{5\sqrt{3}}{3},\ ext{cm}}\n]", "This value is crucial in real-world applications—from architectural design to engineering—where knowing the space available inside curved or polygonal enclosures is essential.", "---", "## Key Takeaways", "- The inradius of an equilateral triangle simplifies to ( \frac{s\sqrt{3}}{6} ).\n- For ( s = 10,\ ext{cm} ), ( r = \frac{5\sqrt{3}}{3},\ ext{cm} \approx 2.89,\ ext{cm} ).\n- This formula leverages the triangle’s uniformity—for any ( s ), multiply by ( \sqrt{3} ), then divide by 6.\n- The incenter (intersection of angle bisectors) coincides with the circumcenter and orthocenter, emphasizing the symmetry of equilateral figures.", "---", "## Why This Matters", "Understanding the inradius helps learners appreciate geometric harmony. Whether constructing a logo, analyzing stress in materials, or designing roundabouts, the inradius provides a precise measure of internal “reach.”", "With side length 10 cm, the inradius of ( \frac{5\sqrt{3}}{3} ) cm is not just a number—it’s a gateway to unlocking spatial reasoning and practical problem-solving.", "---", "Keywords: inradius, equilateral triangle, inscribed circle, incenter radius, formula radius, side length 10 cm, geometric formula, triangle inradius, √3, geometry tips, math education", "---", "Want to calculate your own inradius? Just multiply your equilateral triangle’s side length by ( \sqrt{3} ), then divide by 6—simple, elegant, and powerful."]