A sector of $56^\circ$ has an area of 17.6 cm². The its radius will be

### Concept & Formula To find the radius when the sector area and central angle are given, we use the standard sector area formula and solve for $r$. $$Area = \frac{\theta}{360^\circ} \times \pi r^2$$ ### Step-by-Step Solution * Given Area $= 17.6 \text{ cm}^2$. * Given angle $\theta = 56^\circ$. * $17.6 = \frac{56}{360} \times \frac{22}{7} \times r^2$ * Simplify the terms: $\frac{56}{7} = 8$. * $17.6 = \frac{8}{360} \times 22 \times r^2$ * Simplify further: $\frac{8}{360} = \frac{1}{45}$. * $17.6 = \frac{22}{45} \times r^2$ * Rearrange to solve for $r^2$: $r^2 = \frac{17.6 \times 45}{22}$ * Note that $17.6 / 22 = 0.8$. * $r^2 = 0.8 \times 45 = 36$. * Therefore, $r = 6 \text{ cm}$. ### Exam Strategy & Shortcut When dividing decimal numbers by integers like 22, look for multiples of 11. $176$ is $11 \times 16$, or simply $22 \times 8$. So $17.6 / 22 = 0.8$. Then $0.8 \times 45$ is equivalent to $8 \times 4.5 = 36$. Finding $r = \sqrt{36} = 6$ becomes a quick mental math step. ### Common Pitfall A common pitfall is fumbling the decimal division. If $17.6 / 22$ is tricky, multiply both sides by 10 first to work with integers: $176 = (22/45) \times r^2 \times 10$, making the cancellation straightforward. ### Final Answer Therefore, the correct answer is **6 cm**.