1. **Stating the problem:** Find the perimeter of a sector (පරිමිතිය) given the angle and radius. 2. **Formula for the perimeter of a sector:** $$\text{Perimeter} = 2r + l$$ where $r$ is the radius and $l$ is the arc length. 3. **Arc length formula:** $$l = \frac{\theta}{360^\circ} \times 2\pi r$$ where $\theta$ is the central angle in degrees. 4. **Calculate the perimeter for the first sector:** Given $\theta = 45^\circ$, $r = 7$ cm. Calculate arc length: $$l = \frac{45}{360} \times 2\pi \times 7 = \frac{1}{8} \times 14\pi = \frac{14\pi}{8} = \frac{7\pi}{4}$$ Calculate perimeter: $$\text{Perimeter} = 2 \times 7 + \frac{7\pi}{4} = 14 + \frac{7\pi}{4}$$ 5. **Simplify the perimeter expression:** $$14 + \frac{7\pi}{4} = \frac{56}{4} + \frac{7\pi}{4} = \frac{56 + 7\pi}{4}$$ 6. **Final answer:** The perimeter of the sector with angle $45^\circ$ and radius 7 cm is $$\boxed{\frac{56 + 7\pi}{4} \text{ cm}}$$ This method applies similarly to other sectors by substituting their respective angles and radii.