1. **Problem statement:** We have two concentric circular sectors OKL and OMN with the same central angle of 113°. The smaller sector radius is 15 cm and the larger sector radius is 24 cm. We need to find the perimeter of the shaded shape KLNM, which is the ring-shaped region between the two arcs and the two radial lines. 2. **Formula for arc length:** The length of an arc of a circle is given by $$\text{Arc length} = r \times \theta$$ where $r$ is the radius and $\theta$ is the central angle in radians. 3. **Convert angle to radians:** Since the angle is given in degrees, convert it to radians using $$\theta = 113^\circ \times \frac{\pi}{180} = \frac{113\pi}{180}$$ 4. **Calculate arc lengths:** - Outer arc length (OMN): $$24 \times \frac{113\pi}{180} = \frac{24 \times 113 \pi}{180} = \frac{2712\pi}{180} = 15.0667\pi \approx 47.33\text{ cm}$$ - Inner arc length (OKL): $$15 \times \frac{113\pi}{180} = \frac{15 \times 113 \pi}{180} = \frac{1695\pi}{180} = 9.4167\pi \approx 29.58\text{ cm}$$ 5. **Calculate perimeter of KLNM:** The perimeter consists of the two arcs plus the two radial lines KL and MN. Since KL and MN are radii of the smaller and larger circles respectively, their lengths are 15 cm and 24 cm. So, $$\text{Perimeter} = \text{Outer arc} + \text{Inner arc} + \text{KL} + \text{MN}$$ $$= 47.33 + 29.58 + 15 + 24 = 115.91\text{ cm}$$ 6. **Final answer:** The perimeter of the shaded shape KLNM is $$\boxed{115.91\text{ cm}}$$