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

Let’s solve each problem step by step. We’ll use the standard formulas for spheres and cones.

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Sphere Formulas:
- Volume = (4/3)πr³
- Surface Area = 4πr²

Cone Formulas:
- Volume = (1/3)πr²h
- Curved Surface Area = πrl (where l is slant height)
- Total Surface Area = πrl + πr² (curved + base)

We’ll use π ≈ 3.1416 unless told otherwise, and round to 2 decimal places where needed.

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Problem 1: Sphere with radius 2 cm

Volume = (4/3) × π × (2)³
= (4/3) × π × 8
= (32/3) × π
≈ 10.6667 × 3.1416 ≈ 33.51 cm³

Surface Area = 4 × π × (2)²
= 4 × π × 4
= 16π ≈ 16 × 3.1416 ≈ 50.27 cm²

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Problem 2: Sphere with diameter 14 mm → radius = 7 mm

Volume = (4/3) × π × (7)³
= (4/3) × π × 343
= (1372/3) × π ≈ 457.333 × 3.1416 ≈ 1436.76 mm³

Surface Area = 4 × π × (7)²
= 4 × π × 49
= 196π ≈ 196 × 3.1416 ≈ 615.75 mm²

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Problem 3: Sphere with volume 180 cm³ → find radius

Volume = (4/3)πr³ = 180
→ r³ = 180 × 3 / (4π) = 540 / (4π) = 135 / π
→ r³ ≈ 135 / 3.1416 ≈ 42.97
→ r ≈ ∛42.97 ≈ 3.50 cm (rounded to 2 decimals)

Check: (4/3)π(3.5)³ = (4/3)π(42.875) ≈ (4/3)(134.72) ≈ 179.63 → close enough to 180.

So Radius ≈ 3.50 cm

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Problem 4: Sphere with surface area 25 mm² → find radius

Surface Area = 4πr² = 25
→ r² = 25 / (4π) ≈ 25 / 12.5664 ≈ 1.989
→ r ≈ √1.989 ≈ 1.41 mm

Check: 4π(1.41)² ≈ 4π(1.988) ≈ 25.00 → good.

Radius ≈ 1.41 mm

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Problem 5: Cone with r = 5 cm, h = 12 cm, slant height l = 13 cm

Curved Surface Area = πrl = π × 5 × 13 = 65π ≈ 65 × 3.1416 ≈ 204.20 cm²

Total Surface Area = curved + base = 65π + πr² = 65π + 25π = 90π ≈ 90 × 3.1416 ≈ 282.74 cm²

(Note: You can also calculate base separately: π×5² = 25π ≈ 78.54, then add to 204.20 → same result.)

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Problem 6: Cone with r = 7 cm, slant height l = 25 cm → need height first?

Wait — we’re asked for Volume and Total Surface Area.

But we don’t have vertical height yet. Use Pythagoras:

l² = r² + h² → 25² = 7² + h² → 625 = 49 + h² → h² = 576 → h = 24 cm

Now:

Volume = (1/3)πr²h = (1/3)π×49×24 = (1/3)×1176π = 392π ≈ 392 × 3.1416 ≈ 1231.51 cm³

Total Surface Area = πrl + πr² = π×7×25 + π×49 = 175π + 49π = 224π ≈ 224 × 3.1416 ≈ 703.72 cm²

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Problem 7: Inverted cone? But it says “Volume” and “Total surface area”. Given: diameter = 14.4 cm → radius = 7.2 cm, height = 9.6 cm

This is a cone pointing down, but formulas are same.

Volume = (1/3)πr²h = (1/3)π×(7.2)²×9.6
First, 7.2² = 51.84
Then 51.84 × 9.6 = 497.664
Then ÷3 = 165.888
Then × π ≈ 165.888 × 3.1416 ≈ 521.15 cm³

Now Total Surface Area: Need slant height.

l = √(r² + h²) = √(7.2² + 9.6²) = √(51.84 + 92.16) = √144 = 12 cm

Curved Surface Area = πrl = π×7.2×12 = 86.4π
Base Area = πr² = π×51.84
Total = 86.4π + 51.84π = 138.24π ≈ 138.24 × 3.1416 ≈ 434.29 cm²

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Problem 8: Composite shape — cone on top of hemisphere. Total height = 23 cm, cone height = 15 cm → so hemisphere radius = ?

Hemisphere height = radius → so total height = cone height + hemisphere radius → 23 = 15 + r → r = 8 cm

So both cone and hemisphere have radius 8 cm.

We need Volume and Total Area.

Volume:

Cone volume = (1/3)πr²h = (1/3)π×64×15 = (1/3)×960π = 320π

Hemisphere volume = half of sphere = (1/2)×(4/3)πr³ = (2/3)π×512 = (1024/3)π ≈ 341.333π

Total Volume = 320π + 341.333π = 661.333π ≈ 661.333 × 3.1416 ≈ 2077.68 cm³

Total Area:

Note: The base of the cone is attached to the hemisphere, so we do NOT include the circular base of the cone or the flat face of the hemisphere.

We include:
- Curved surface of cone: πrl
- Curved surface of hemisphere: 2πr² (since full sphere is 4πr², half is 2πr²)

Need slant height of cone: l = √(r² + h²) = √(64 + 225) = √289 = 17 cm

Curved cone area = π×8×17 = 136π

Hemisphere curved area = 2π×64 = 128π

Total Area = 136π + 128π = 264π ≈ 264 × 3.1416 ≈ 829.38 cm²

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Final Answers:

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 = 1.41 mm
5) Curved surface area = 204.20 cm², Total surface area = 282.74 cm²
6) Volume = 1231.51 cm³, Total surface area = 703.72 cm²
7) Volume = 521.15 cm³, Total surface area = 434.29 cm²
8) Volume = 2077.68 cm³, Total area = 829.38 cm²

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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 = 1.41 mm
5) Curved surface area = 204.20 cm², Total surface area = 282.74 cm²
6) Volume = 1231.51 cm³, Total surface area = 703.72 cm²
7) Volume = 521.15 cm³, Total surface area = 434.29 cm²
8) Volume = 2077.68 cm³, Total area = 829.38 cm²