Surface Area Playground 03F9A1

1. **Problem Statement:** We have three solid shapes: a cube, a cone, and a cylinder. Given dimensions: - Cube side length $s = 125$ cm - Cone slant height $l = 135$ cm, base radius $r = 62.5$ cm (since cone sits on cube of side 125 cm, radius is half side) - Cylinder radius $r = 62.5$ cm, height $h = 125$ cm We need to: - a) Draw nets for each shape with measurements. - b) Calculate surface area of each shape before assembly. --- 2. **Nets and Measurements:** - Cube net: 6 squares each of side $125$ cm. - Cone net: a circular base of radius $62.5$ cm and a sector of a circle with radius $135$ cm (slant height) and arc length equal to circumference of base $2\pi \times 62.5$ cm. - Cylinder net: two circles of radius $62.5$ cm and a rectangle of height $125$ cm and width equal to circumference $2\pi \times 62.5$ cm. --- 3. **Surface Area Formulas:** - Cube surface area: $SA_{cube} = 6s^2$ - Cone surface area (excluding base): $SA_{cone} = \pi r l$ - Cylinder surface area (excluding bases): $SA_{cyl\,side} = 2\pi r h$ - Cylinder total surface area: $SA_{cyl} = 2\pi r^2 + 2\pi r h$ --- 4. **Calculations:** **Cube:** $$SA_{cube} = 6 \times 125^2 = 6 \times 15625 = 93750 \text{ cm}^2$$ **Cone:** Base radius $r = \frac{125}{2} = 62.5$ cm Slant height $l = 135$ cm Lateral surface area: $$SA_{cone} = \pi \times 62.5 \times 135 = 8437.5\pi \approx 26490.82 \text{ cm}^2$$ Base area: $$A_{base} = \pi r^2 = \pi \times 62.5^2 = 3906.25\pi \approx 12265.36 \text{ cm}^2$$ Total cone surface area: $$SA_{cone,total} = 26490.82 + 12265.36 = 38756.18 \text{ cm}^2$$ **Cylinder:** Radius $r = 62.5$ cm, height $h = 125$ cm Lateral surface area: $$SA_{cyl,side} = 2\pi \times 62.5 \times 125 = 15625\pi \approx 49087.39 \text{ cm}^2$$ Base area (two bases): $$2 \times \pi r^2 = 2 \times \pi \times 62.5^2 = 2 \times 3906.25\pi = 7812.5\pi \approx 24530.72 \text{ cm}^2$$ Total cylinder surface area: $$SA_{cyl,total} = 49087.39 + 24530.72 = 73618.11 \text{ cm}^2$$ --- 5. **Summary of Surface Areas (rounded to nearest hundredth):** - Cube: $93750.00$ cm$^2$ - Cone: $38756.18$ cm$^2$ - Cylinder: $73618.11$ cm$^2$ These are the surface areas before assembly.