The radius of a circle is 20% more than the height of a right-angled triangle. The base of the triangle is 36 cm. If the area of triangle and circle be equal, what will be the area of circle?

### Concept & Equation Equate the area formula of the right-angled triangle with the area formula of the circle, substituting the known relationship between the radius and the height. $$\text{Area}_{\text{triangle}} = \frac{1}{2} \times b \times h$$ $$\text{Area}_{\text{circle}} = \pi r^2$$ ### Step-by-Step Solution * Given: Base of the triangle $b = 36$ cm. * Let the height of the triangle be $h$. * The radius $r$ is 20% more than $h$: $r = h + 0.20h = 1.2h = \frac{6}{5}h$. * Set the areas equal: $\frac{1}{2} \times 36 \times h = \pi \times \left(\frac{6}{5}h\right)^2$ $18h = \pi \times \frac{36}{25}h^2$. * Divide by $h$ (since $h > 0$): $18 = \pi \times \frac{36}{25}h$ $h = \frac{18 \times 25}{36\pi} = \frac{25}{2\pi}$. * The area of the circle is equal to the area of the triangle: $\text{Area} = 18h = 18 \times \left(\frac{25}{2\pi}\right) = \frac{225}{\pi}$. * Because $\pi$ is an irrational number ($\approx 3.14159$), $\frac{225}{\pi}$ will not yield an exact integer value matching any of the specific numerical options provided (72, 128, 144, 216). ### Exam Strategy & Shortcut Observe the final equivalence $18h = \pi r^2$. Since $r = 1.2h$, the $\pi$ term cannot be eliminated from the final calculation for $h$ or the Area. Seeing that all numerical options are clean integers, you can quickly deduce that a definitive integer answer cannot be determined without an approximation for $\pi$ being explicitly stated in the problem (which it is not). ### Common Pitfall Approximating $\pi$ as $3$ to force a clean integer result, leading to a mathematically unsound selection of one of the incorrect distractor options. ### Final Answer Therefore, the correct answer is **Cannot be determined**.