The sides of a triangle are 6 cm, 11 cm and 15 cm. The radius of its incircle is

### Concept & Inradius Formula The radius of an incircle ($r$) of any triangle can be found by dividing the area of the triangle ($A$) by its semi-perimeter ($s$). $$r = \frac{A}{s}$$ The area $A$ is found using Heron's formula: $$A = \sqrt{s(s-a)(s-b)(s-c)}$$ ### Step-by-Step Solution 1. **Calculate the Semi-perimeter ($s$):** $s = \frac{a + b + c}{2} = \frac{6 + 11 + 15}{2} = \frac{32}{2} = 16$ cm. 2. **Calculate the Area ($A$) using Heron's Formula:** $A = \sqrt{16(16 - 6)(16 - 11)(16 - 15)}$ $A = \sqrt{16 \times 10 \times 5 \times 1}$ $A = \sqrt{800} = \sqrt{400 \times 2} = 20\sqrt{2}$ cm$^2$. 3. **Calculate the Inradius ($r$):** $r = \frac{A}{s} = \frac{20\sqrt{2}}{16}$ Simplify the fraction by dividing the numerator and denominator by 4: $r = \frac{5\sqrt{2}}{4}$ cm. ### Exam Strategy & Shortcut To compute $\sqrt{800}$ quickly, group the factors logically: $16 \times (10 \times 5) = 16 \times 50 = 800$. You know $\sqrt{16} = 4$ and $\sqrt{50} = 5\sqrt{2}$. Multiply them directly: $4 \times 5\sqrt{2} = 20\sqrt{2}$. This saves time factoring large numbers. ### Common Pitfall A frequent error is forgetting to divide the area by the semi-perimeter $s$ and instead dividing by the full perimeter, which would yield exactly half of the correct answer. Always remember $r = A/s$, not $A/P$. ### Final Answer Therefore, the correct answer is **$\frac{5\sqrt{2}}{4}$ cm**.