What will be the area of a semi-circle whose perimeter is 36 cm?

### Concept & Perimeter to Area Relationship for Semi-circle To find the area, you must first extract the radius from the given perimeter. The perimeter of a semi-circle includes both the arc and the diameter. $$ \text{Perimeter} = \pi r + 2r = r\left(\frac{36}{7}\right) $$ $$ \text{Area} = \frac{1}{2} \pi r^2 $$ ### Step-by-Step Solution * **Set up the perimeter equation:** $P = \pi r + 2r = 36$ $r\left(\frac{22}{7} + 2\right) = 36$ $r\left(\frac{22 + 14}{7}\right) = 36$ $r\left(\frac{36}{7}\right) = 36$ * **Solve for radius:** $r = 36 \times \frac{7}{36} = 7$ cm. * **Calculate the area:** $\text{Area} = \frac{1}{2} \pi r^2 = \frac{1}{2} \times \frac{22}{7} \times (7)^2$ $\text{Area} = \frac{1}{2} \times 22 \times 7 = 11 \times 7 = 77$ cm$^2$. * **Check the options:** The calculated area is 77 cm$^2$, which is not listed in options (a), (b), or (c). ### Exam Strategy & Shortcut Memorize the specific fractional multiplier for a semi-circle's perimeter: $P = \frac{36}{7} r$. If $P = 36$, it immediately implies $r = 7$. From standard circle values, a radius of 7 means a full circle area of 154, so a semi-circle is 77. The answer is clearly missing. ### Common Pitfall A standard error is mistaking the formula for the perimeter of a semi-circle as just $\pi r$ (which would give $r = \frac{36 \times 7}{22} \approx 11.45$ and lead to messy, incorrect calculations). ### Final Answer Therefore, the correct answer is **None of these**.