Math worksheet for calculating volume and surface area of cones and spheres.

I'll solve each problem step by step, working through all the calculations carefully.

Problem 1: Sphere with radius 2 cm
- Volume of sphere = (4/3)πr³
- Volume = (4/3) × π × 2³ = (4/3) × π × 8 = 32π/3 ≈ 33.51 cm³
- Surface area of sphere = 4πr²
- Surface area = 4 × π × 2² = 4 × π × 4 = 16π ≈ 50.27 cm²

Problem 2: Sphere with diameter 14 mm (radius = 7 mm)
- Volume = (4/3)πr³ = (4/3) × π × 7³ = (4/3) × π × 343 = 1372π/3 ≈ 1436.76 mm³
- Surface area = 4πr² = 4 × π × 7² = 4 × π × 49 = 196π ≈ 615.75 mm²

Problem 3: Sphere with volume 180 cm³
- Volume = (4/3)πr³ = 180
- r³ = 180 × 3/(4π) = 540/(4π) = 135/π
- r³ ≈ 42.97
- r ≈ ∛42.97 ≈ 3.50 cm

Problem 4: Hemisphere with surface area 25 mm²
- Surface area of hemisphere = 3πr² (includes curved surface + base)
- 3πr² = 25
- r² = 25/(3π) ≈ 2.65
- r ≈ √2.65 ≈ 1.63 mm

Problem 5: Cone with height 12 cm, slant height 13 cm, radius 5 cm
- Curved surface area = πrl = π × 5 × 13 = 65π ≈ 204.20 cm²
- Total surface area = πrl + πr² = 65π + π(5²) = 65π + 25π = 90π ≈ 282.74 cm²

Problem 6: Cone with radius 7 cm, slant height 25 cm
- First find height using Pythagoras: h² + r² = l²
- h² + 7² = 25²
- h² + 49 = 625
- h² = 576
- h = 24 cm
- Volume = (1/3)πr²h = (1/3) × π × 7² × 24 = (1/3) × π × 49 × 24 = 392π ≈ 1231.50 cm³
- Total surface area = πrl + πr² = π × 7 × 25 + π × 7² = 175π + 49π = 224π ≈ 703.72 cm²

Problem 7: Inverted cone with diameter 14.4 cm (radius = 7.2 cm), height 9.6 cm
- First find slant height: l² = r² + h² = 7.2² + 9.6² = 51.84 + 92.16 = 144
- l = 12 cm
- Volume = (1/3)πr²h = (1/3) × π × 7.2² × 9.6 = (1/3) × π × 51.84 × 9.6 = 165.89π ≈ 521.15 cm³
- Total surface area = πrl + πr² = π × 7.2 × 12 + π × 7.2² = 86.4π + 51.84π = 138.24π ≈ 434.29 cm²

Problem 8: Cone on top of hemisphere
- Hemisphere radius = 15 cm
- Cone height = 23 - 15 = 8 cm
- Cone radius = 15 cm
- Slant height of cone: l² = 15² + 8² = 225 + 64 = 289
- l = 17 cm
- Volume of cone = (1/3)πr²h = (1/3) × π × 15² × 8 = (1/3) × π × 225 × 8 = 600π
- Volume of hemisphere = (2/3)πr³ = (2/3) × π × 15³ = (2/3) × π × 3375 = 2250π
- Total volume = 600π + 2250π = 2850π ≈ 8953.54 cm³
- Total area = curved surface of cone + curved surface of hemisphere
- Curved surface of cone = πrl = π × 15 × 17 = 255π
- Curved surface of hemisphere = 2πr² = 2 × π × 15² = 450π
- Total area = 255π + 450π = 705π ≈ 2214.82 cm²

Final Answer:
1) Volume = 33.51 cm³, Surface area = 50.27 cm²
2) Volume = 1436.76 mm³, Surface area = 615.75 mm²
3) Radius = 3.50 cm
4) Radius = 1.63 mm
5) Curved surface area = 204.20 cm², Total surface area = 282.74 cm²
6) Volume = 1231.50 cm³, Total surface area = 703.72 cm²
7) Volume = 521.15 cm³, Total surface area = 434.29 cm²
8) Volume = 8953.54 cm³, Total area = 2214.82 cm²