1. **Problem statement:** Calculate the volume (V), lateral surface area (LSA), and total surface area (TSA) of a cylinder with diameter $d=10$ cm and height $h=12$ cm. 2. **Formulas:** - Radius $r = \frac{d}{2}$ - Volume of cylinder: $$V = \pi r^2 h$$ - Lateral surface area: $$LSA = 2 \pi r h$$ - Total surface area: $$TSA = LSA + 2 \pi r^2$$ (includes top and bottom) 3. **Calculate radius:** $$r = \frac{10}{2} = 5\text{ cm}$$ 4. **Calculate volume:** $$V = \pi (5)^2 (12) = \pi \times 25 \times 12 = 300\pi \approx 942.48\text{ cm}^3$$ 5. **Calculate lateral surface area:** $$LSA = 2 \pi (5)(12) = 120\pi \approx 376.99\text{ cm}^2$$ 6. **Calculate total surface area:** $$TSA = 120\pi + 2 \pi (5)^2 = 120\pi + 50\pi = 170\pi \approx 533.52\text{ cm}^2$$ **Final answers:** - Volume $V \approx 942.48$ cm³ - Lateral surface area $LSA \approx 376.99$ cm² - Total surface area $TSA \approx 533.52$ cm²