Volume and Surface Area of Cones and Spheres Worksheet | PDF ... - Free Printable

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

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

Cone:
- Volume = (1/3)πr²h
- Curved Surface Area = πrl (where l is slant height)
- Total Surface Area = πr(l + r)

We’ll use π ≈ 3.1416, and round answers 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 ≈ 42.9718
→ r ≈ ∛42.9718 ≈ 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.

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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

Check: 4π(1.41)² ≈ 4π(1.9881) ≈ 25.00 — good.

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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 = πr(l + r) = π × 5 × (13 + 5) = π × 5 × 18 = 90π ≈ 90 × 3.1416 ≈ 282.74 cm²

(Note: You can also add base area πr² = 25π to curved surface 65π → total 90π — 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 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)(1176)π = 392π ≈ 392 × 3.1416 ≈ 1231.51 cm³

Total Surface Area = πr(l + r) = π × 7 × (25 + 7) = π × 7 × 32 = 224π ≈ 224 × 3.1416 ≈ 703.72 cm²

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Problem 7: Inverted cone? But it says “cone” — looks like a 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)

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

Total Surface Area: Need slant height l.

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

Total Surface Area = πr(l + r) = π × 7.2 × (12 + 7.2) = π × 7.2 × 19.2

7.2 × 19.2 = let’s compute: 7×19.2=134.4, 0.2×19.2=3.84 → total 138.24

So 138.24π ≈ 138.24 × 3.1416 ≈ 434.29 cm²

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Problem 8: Composite shape — cone on top of hemisphere.

Given:
- Total height from tip to bottom of hemisphere = 23 cm
- Height of cone part = 15 cm → so radius of hemisphere = 23 - 15 = 8 cm? Wait — no.

Actually, look at diagram: The 15 cm is labeled as the height of the cone (from apex to base). The 23 cm is total height from apex to bottom of hemisphere. So the hemisphere has radius = 23 - 15 = 8 cm.

Also, since the cone sits on the hemisphere, they share the same radius → r = 8 cm.

So:

Volume of whole shape = volume of cone + volume of hemisphere

Cone: r = 8 cm, h = 15 cm
Volume_cone = (1/3)πr²h = (1/3)π(64)(15) = (1/3)(960)π = 320π

Hemisphere: half of sphere → (2/3)πr³ = (2/3)π(512) = (1024/3)π ≈ 341.333π

Total Volume = 320π + 341.333π = 661.333π ≈ 661.333 × 3.1416 ≈ 2077.58 cm³

Total Area: This means external surface area only — not including the circular base where cone and hemisphere join (since it’s internal).

So:
- Curved surface area of cone: πrl → need slant height l.

l = √(r² + h²) = √(64 + 225) = √289 = 17 cm

Curved SA of cone = π × 8 × 17 = 136π

Curved SA of hemisphere = half of sphere’s surface area = (1/2)(4πr²) = 2πr² = 2π(64) = 128π

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

Note: We do NOT include the flat circular base of the hemisphere because it’s sitting on something? Actually, in composite shapes like this, unless specified, we usually include all outer surfaces. But here, the base of the hemisphere is exposed? Looking at diagram — it shows the hemisphere at the bottom, so its flat face is probably attached to nothing — but wait, the label says “hemisphere”, which typically includes the curved part only when combined. However, in many problems, for a solid made of cone + hemisphere, “total area” means curved surface of cone + curved surface of hemisphere (excluding the joining circle).

Yes — standard practice: exclude the interface. So our calculation above is correct: 136π + 128π = 264π.

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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 = 2077.58 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 = 2077.58 cm³, Total area = 829.38 cm²