1. **Problem statement:** Calculate the volume and surface area of the following solids with given dimensions: - Cube with edge $a=3.5$ cm - Cuboid with edges $a=3.5$ cm, $b=4.6$ cm, $c=5.2$ cm - Cylinder with radius $r=4$ cm and height $h=6$ cm - Pyramid with square base edge $a=4$ cm and height $h=8$ cm - Cone with radius $r=3$ cm and height $h=5$ cm - Sphere with radius $r=5$ cm 2. **Formulas and rules:** - Cube volume: $V = a^3$ - Cube surface area: $O = 6a^2$ - Cuboid volume: $V = abc$ - Cuboid surface area: $O = 2(ab + bc + ac)$ - Cylinder volume: $V = \pi r^2 h$ - Cylinder surface area: $O = 2\pi r (r + h)$ - Pyramid volume: $V = \frac{1}{3} a^2 h$ - Pyramid surface area: $O = a^2 + 2a s$ where $s = \sqrt{\left(\frac{a}{2}\right)^2 + h^2}$ (slant height) - Cone volume: $V = \frac{1}{3} \pi r^2 h$ - Cone surface area: $O = \pi r (r + s)$ where $s = \sqrt{r^2 + h^2}$ (slant height) - Sphere volume: $V = \frac{4}{3} \pi r^3$ - Sphere surface area: $O = 4 \pi r^2$ 3. **Calculations:** **Cube:** $$V = 3.5^3 = 42.875 \text{ cm}^3$$ $$O = 6 \times 3.5^2 = 6 \times 12.25 = 73.5 \text{ cm}^2$$ **Cuboid:** $$V = 3.5 \times 4.6 \times 5.2 = 83.72 \text{ cm}^3$$ $$O = 2(3.5 \times 4.6 + 4.6 \times 5.2 + 3.5 \times 5.2)$$ $$= 2(16.1 + 23.92 + 18.2) = 2 \times 58.22 = 116.44 \text{ cm}^2$$ **Cylinder:** $$V = \pi \times 4^2 \times 6 = \pi \times 16 \times 6 = 96\pi \approx 301.59 \text{ cm}^3$$ $$O = 2\pi \times 4 (4 + 6) = 8\pi \times 10 = 80\pi \approx 251.33 \text{ cm}^2$$ **Pyramid:** Calculate slant height: $$s = \sqrt{\left(\frac{4}{2}\right)^2 + 8^2} = \sqrt{2^2 + 64} = \sqrt{68} \approx 8.246$$ Volume: $$V = \frac{1}{3} \times 4^2 \times 8 = \frac{1}{3} \times 16 \times 8 = \frac{128}{3} \approx 42.67 \text{ cm}^3$$ Surface area: $$O = 16 + 2 \times 4 \times 8.246 = 16 + 65.97 = 81.97 \text{ cm}^2$$ **Cone:** Calculate slant height: $$s = \sqrt{3^2 + 5^2} = \sqrt{9 + 25} = \sqrt{34} \approx 5.831$$ Volume: $$V = \frac{1}{3} \pi \times 3^2 \times 5 = \frac{1}{3} \pi \times 9 \times 5 = 15\pi \approx 47.12 \text{ cm}^3$$ Surface area: $$O = \pi \times 3 (3 + 5.831) = 3\pi \times 8.831 = 26.49\pi \approx 83.23 \text{ cm}^2$$ **Sphere:** Volume: $$V = \frac{4}{3} \pi \times 5^3 = \frac{4}{3} \pi \times 125 = \frac{500}{3} \pi \approx 523.6 \text{ cm}^3$$ Surface area: $$O = 4 \pi \times 5^2 = 4 \pi \times 25 = 100\pi \approx 314.16 \text{ cm}^2$$ 4. **Final answers:** - Cube: $V=42.875$ cm$^3$, $O=73.5$ cm$^2$ - Cuboid: $V=83.72$ cm$^3$, $O=116.44$ cm$^2$ - Cylinder: $V\approx301.59$ cm$^3$, $O\approx251.33$ cm$^2$ - Pyramid: $V\approx42.67$ cm$^3$, $O\approx81.97$ cm$^2$ - Cone: $V\approx47.12$ cm$^3$, $O\approx83.23$ cm$^2$ - Sphere: $V\approx523.6$ cm$^3$, $O\approx314.16$ cm$^2$