The circumradius of an equilateral triangle is 8 cm. The inradius of the triangle

### Concept & Inradius and Circumradius of Equilateral Triangle For an equilateral triangle, the centroid, circumcenter, and incenter all coincide at the same point. This center divides the median in the ratio $2 : 1$. The larger segment (vertex to center) is the circumradius $R$, and the smaller segment (center to side) is the inradius $r$. $$ R = 2r $$ ### Step-by-Step Solution * Given: Circumradius $R = 8$ cm. * We know the relationship between circumradius and inradius in an equilateral triangle is $R = 2r$. * Substituting the given value: $8 = 2r$. * Solving for $r$: $r = \frac{8}{2} = 4$ cm. ### Exam Strategy & Shortcut This is a direct property question. Memorize the ratio $R : r = 2 : 1$ for equilateral triangles. If $R$ is 8, $r$ is instantly half of that, which is 4. No side length calculations are needed. ### Common Pitfall A common mistake is attempting to calculate the side of the triangle first using $R = \frac{a}{\sqrt{3}}$ and then plugging it into $r = \frac{a}{2\sqrt{3}}$. While mathematically sound, it is a massive waste of time in a competitive exam. ### Final Answer Therefore, the correct answer is **4 cm**.