1. The problem is to find the volume or surface area of a quarter cylinder, which is one-fourth of a full cylinder. 2. Recall the formulas for a full cylinder: - Volume: $$V = \pi r^2 h$$ - Surface area (including top and bottom): $$A = 2\pi r h + 2\pi r^2$$ 3. Since a quarter cylinder is exactly one-fourth of a full cylinder, we divide these formulas by 4: - Quarter cylinder volume: $$V_{quarter} = \frac{1}{4} \pi r^2 h$$ - Quarter cylinder surface area is more complex because it includes two flat rectangular faces (the quarter cuts) plus the curved surface and one circular quarter of the base and top. 4. The curved surface area of the quarter cylinder is one-fourth of the full curved surface: $$A_{curved} = \frac{1}{4} (2\pi r h) = \frac{\pi r h}{2}$$ 5. The two flat rectangular faces formed by the quarter cut each have area $$r \times h$$, so total flat face area: $$A_{flat} = 2 r h$$ 6. The quarter circular base and top each have area $$\frac{1}{4} \pi r^2$$, so total base and top area: $$A_{base+top} = 2 \times \frac{1}{4} \pi r^2 = \frac{\pi r^2}{2}$$ 7. Total surface area of the quarter cylinder is the sum: $$A_{total} = A_{curved} + A_{flat} + A_{base+top} = \frac{\pi r h}{2} + 2 r h + \frac{\pi r^2}{2}$$ This formula accounts for all surfaces of the quarter cylinder. Final answers: - Volume: $$V = \frac{1}{4} \pi r^2 h$$ - Surface area: $$A = \frac{\pi r h}{2} + 2 r h + \frac{\pi r^2}{2}$$