The perimeter of a triangle is 30 cm and the circumference of its incircle is 88 cm. The area of the triangle is

### Concept & Inradius and Area Relationship The area ($A$) of any triangle can be expressed as the product of its inradius ($r$) and its semi-perimeter ($s$): $$A = r \times s$$ The circumference of the incircle is $C = 2\pi r$, which allows us to find the inradius $r$. ### Step-by-Step Solution 1. **Find the Inradius ($r$):** Given the circumference is 88 cm: $2\pi r = 88$ $2 \times \frac{22}{7} \times r = 88$ $\frac{44}{7} \times r = 88$ $r = \frac{88 \times 7}{44} = 2 \times 7 = 14$ cm. 2. **Find the Semi-perimeter ($s$):** Given the perimeter is 30 cm: $s = \frac{\text{Perimeter}}{2} = \frac{30}{2} = 15$ cm. 3. **Calculate the Area ($A$):** $A = r \times s$ $A = 14 \times 15 = 210$ cm$^2$. ### Exam Strategy & Shortcut When you see a circumference of 88, recognize instantly that it's double 44, which corresponds to a radius of 14 (since $C=44$ means $r=7$ for $\pi=22/7$). Then immediately multiply this radius (14) by half the given perimeter (15) to get $14 \times 15 = 210$. The entire process can be done mentally. ### Common Pitfall Students often multiply the inradius by the full perimeter instead of the semi-perimeter, which leads to the incorrect answer of 420 cm$^2$. Always remember $A = rs$, not $rp$. ### Final Answer Therefore, the correct answer is **210 cm$^2**.