1. **State the problem:** Calculate the perimeter of the given circular sectors with specified angles and radii. 2. **Formula for perimeter of a circular sector:** $$\text{Perimeter} = 2r + s$$ where $r$ is the radius and $s$ is the arc length. 3. **Arc length formula:** $$s = r \theta$$ where $\theta$ is the angle in radians. 4. **Convert degrees to radians:** $$\theta = \frac{\pi}{180} \times \text{angle in degrees}$$ 5. **Calculate each perimeter:** **i.** $45^\circ$, $r=7$ cm $$\theta = \frac{\pi}{180} \times 45 = \frac{\pi}{4}$$ $$s = 7 \times \frac{\pi}{4} = \frac{7\pi}{4}$$ $$\text{Perimeter} = 2 \times 7 + \frac{7\pi}{4} = 14 + \frac{7\pi}{4} \approx 14 + 5.50 = 19.50 \text{ cm}$$ **ii.** $120^\circ$, $r=21$ cm $$\theta = \frac{\pi}{180} \times 120 = \frac{2\pi}{3}$$ $$s = 21 \times \frac{2\pi}{3} = 14\pi$$ $$\text{Perimeter} = 2 \times 21 + 14\pi = 42 + 14\pi \approx 42 + 43.98 = 85.98 \text{ cm}$$ **iii.** $r=21$ cm, right angle sector means $90^\circ$ $$\theta = \frac{\pi}{180} \times 90 = \frac{\pi}{2}$$ $$s = 21 \times \frac{\pi}{2} = \frac{21\pi}{2}$$ $$\text{Perimeter} = 2 \times 21 + \frac{21\pi}{2} = 42 + 10.50\pi \approx 42 + 32.99 = 74.99 \text{ cm}$$ **iv.** $60^\circ$, $r=35$ cm $$\theta = \frac{\pi}{180} \times 60 = \frac{\pi}{3}$$ $$s = 35 \times \frac{\pi}{3} = \frac{35\pi}{3}$$ $$\text{Perimeter} = 2 \times 35 + \frac{35\pi}{3} = 70 + 11.67\pi \approx 70 + 36.65 = 106.65 \text{ cm}$$ **v.** $30^\circ$, $r=14$ cm $$\theta = \frac{\pi}{180} \times 30 = \frac{\pi}{6}$$ $$s = 14 \times \frac{\pi}{6} = \frac{14\pi}{6} = \frac{7\pi}{3}$$ $$\text{Perimeter} = 2 \times 14 + \frac{7\pi}{3} = 28 + 7.33\pi \approx 28 + 22.99 = 50.99 \text{ cm}$$ **vi.** $240^\circ$, $r=10.5$ cm $$\theta = \frac{\pi}{180} \times 240 = \frac{4\pi}{3}$$ $$s = 10.5 \times \frac{4\pi}{3} = 14\pi$$ $$\text{Perimeter} = 2 \times 10.5 + 14\pi = 21 + 14\pi \approx 21 + 43.98 = 64.98 \text{ cm}$$ **Final answers:** 1. $19.50$ cm 2. $85.98$ cm 3. $74.99$ cm 4. $106.65$ cm 5. $50.99$ cm 6. $64.98$ cm