1. **Problem statement:** We have a circle with center O and radius 6 cm. Two radii OA and OD form an angle of 62°. Points B and C lie on the circle such that AB = DC = 5 cm outside the circle. The shape is formed by a minor sector OBC and a major sector OAD joined along AB and DC. We need to find (a) the area of the logo design and (b) the perimeter of the logo design. 2. **Understanding the sectors:** - The circle radius is $r = 6$ cm. - The angle of the major sector $\angle AOD = 62^\circ$. - The minor sector OBC corresponds to the remaining angle $360^\circ - 62^\circ = 298^\circ$. 3. **Formula for sector area:** The area of a sector with radius $r$ and angle $\theta$ (in degrees) is $$\text{Area} = \frac{\theta}{360} \times \pi r^2$$ 4. **Calculate areas of sectors:** - Area of major sector OAD: $$A_{OAD} = \frac{62}{360} \times \pi \times 6^2 = \frac{62}{360} \times \pi \times 36 = \frac{62 \times 36}{360} \pi = 6.2\pi \approx 19.47 \text{ cm}^2$$ - Area of minor sector OBC: $$A_{OBC} = \frac{298}{360} \times \pi \times 6^2 = \frac{298}{360} \times \pi \times 36 = \frac{298 \times 36}{360} \pi = 29.8\pi \approx 93.62 \text{ cm}^2$$ 5. **Calculate total area of the logo design:** Since the logo is formed by joining these two sectors along AB and DC, the total area is the sum of the two sector areas: $$A_{total} = A_{OAD} + A_{OBC} = 6.2\pi + 29.8\pi = 36\pi \approx 113.1 \text{ cm}^2$$ 6. **Calculate the perimeter of the logo design:** The perimeter consists of: - The two chords AB and DC, each 5 cm. - The two arcs of the sectors OAD and OBC. 7. **Length of arcs:** Arc length formula: $$L = \frac{\theta}{360} \times 2\pi r$$ - Arc length of major sector OAD: $$L_{OAD} = \frac{62}{360} \times 2\pi \times 6 = \frac{62}{360} \times 12\pi = 2.0667\pi \approx 6.49 \text{ cm}$$ - Arc length of minor sector OBC: $$L_{OBC} = \frac{298}{360} \times 2\pi \times 6 = \frac{298}{360} \times 12\pi = 9.9333\pi \approx 31.21 \text{ cm}$$ 8. **Total perimeter:** $$P = AB + DC + L_{OAD} + L_{OBC} = 5 + 5 + 6.49 + 31.21 = 47.7 \text{ cm}$$ **Final answers:** - (a) Area of the logo design $\approx 113.1$ cm$^2$ - (b) Perimeter of the logo design $\approx 47.7$ cm