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

Let’s solve each problem step by step.

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Problem 1: Sphere with radius = 2 cm

Volume of sphere = (4/3)πr³
= (4/3) × π × (2)³
= (4/3) × π × 8
= 32π/3 ≈ 33.51 cm³

Surface area of sphere = 4πr²
= 4 × π × (2)²
= 4 × π × 4
= 16π ≈ 50.27 cm²

✔ Volume = 33.51 cm³
✔ Surface area = 50.27 cm²

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

Volume = (4/3)πr³
= (4/3) × π × (7)³
= (4/3) × π × 343
= 1372π/3 ≈ 1436.76 mm³

Surface area = 4πr²
= 4 × π × 49
= 196π ≈ 615.75 mm²

✔ Volume = 1436.76 mm³
✔ Surface area = 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

✔ 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 ≈ 1.9894
→ r ≈ √1.9894 ≈ 1.41 mm

✔ Radius = 1.41 mm

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Problem 5: Cone with radius = 5 cm, height = 12 cm, slant height = 13 cm

Curved surface area = πrl = π × 5 × 13 = 65π ≈ 204.20 cm²

Total surface area = curved + base = πrl + πr² = 65π + 25π = 90π ≈ 282.74 cm²

✔ Curved surface area = 204.20 cm²
✔ Total surface area = 282.74 cm²

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Problem 6: Cone with radius = 7 cm, slant height = 25 cm → find volume and total surface area

First, find height using Pythagoras:
h = √(l² - r²) = √(25² - 7²) = √(625 - 49) = √576 = 24 cm

Volume = (1/3)πr²h = (1/3) × π × 49 × 24 = (1/3) × 1176π = 392π ≈ 1231.50 cm³

Total surface area = πr² + πrl = π×49 + π×7×25 = 49π + 175π = 224π ≈ 703.72 cm²

✔ Volume = 1231.50 cm³
✔ Total surface area = 703.72 cm²

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Problem 7: Cone with diameter = 14.4 cm → radius = 7.2 cm, height = 9.6 cm

Volume = (1/3)πr²h = (1/3) × π × (7.2)² × 9.6
= (1/3) × π × 51.84 × 9.6
= (1/3) × π × 497.664
= 165.888π ≈ 521.15 cm³

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

Total surface area = πr² + πrl = π×51.84 + π×7.2×12 = 51.84π + 86.4π = 138.24π ≈ 434.30 cm²

✔ Volume = 521.15 cm³
✔ Total surface area = 434.30 cm²

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Problem 8: Composite shape — cone on top of hemisphere

Cone: height = 15 cm, radius = ? Wait — diagram shows total height from bottom to tip is 23 cm, and cone height is 15 cm → so hemisphere radius = 23 - 15 = 8 cm? But wait — let’s check labels.

Actually, looking at diagram:
- The cone has height 15 cm (from its base to tip).
- The hemisphere is below it, and the total height from bottom of hemisphere to tip of cone is 23 cm → so hemisphere radius = 23 - 15 = 8 cm.
BUT — the label “15 cm” is pointing to the cone’s height, and “23 cm” is total height. So yes, hemisphere radius = 8 cm.

Wait — actually, re-examining: the diagram says “Cone” with height 15 cm, and “Hemisphere” below it, and total height 23 cm → so hemisphere radius = 23 - 15 = 8 cm. But that would mean the cone’s base radius is also 8 cm (since they’re joined).

So:

Cone: r = 8 cm, h = 15 cm
Hemisphere: r = 8 cm

Volume of cone = (1/3)πr²h = (1/3)π×64×15 = 320π ≈ 1005.31 cm³
Volume of hemisphere = (2/3)πr³ = (2/3)π×512 = 1024π/3 ≈ 1072.33 cm³
Total volume = 320π + 1024π/3 = (960π + 1024π)/3 = 1984π/3 ≈ 2077.64 cm³

Total area = curved surface of cone + curved surface of hemisphere (no flat circles since they’re joined)

Curved surface of cone = πrl
l = √(r² + h²) = √(64 + 225) = √289 = 17 cm
→ π×8×17 = 136π

Curved surface of hemisphere = 2πr² = 2π×64 = 128π

Total area = 136π + 128π = 264π ≈ 829.38 cm²

✔ Volume = 2077.64 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 = 1231.50 cm³, Total surface area = 703.72 cm²
7) Volume = 521.15 cm³, Total surface area = 434.30 cm²
8) Volume = 2077.64 cm³, Total area = 829.38 cm²