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

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

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Sphere Formulas:

- Volume = (4/3) × π × r³
- Surface Area = 4 × π × r²

Cone Formulas:

- Volume = (1/3) × π × r² × h
- Curved Surface Area = π × r × l (where l is slant height)
- Total Surface Area = π × r × l + π × r² (curved + base)

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

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Problem 1: Sphere with radius 2 cm

Volume = (4/3) × π × (2)³ = (4/3) × π × 8 = (32/3)π ≈ 33.51 cm³
Surface Area = 4 × π × (2)² = 4 × π × 4 = 16π ≈ 50.27 cm²

✔ Answer: 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) × π × (7)³ = (4/3) × π × 343 = (1372/3)π ≈ 1436.76 mm³
Surface Area = 4 × π × (7)² = 4 × π × 49 = 196π ≈ 615.75 mm²

✔ Answer: 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 ≈ 135 / 3.1416 ≈ 42.97
→ r = ∛42.97 ≈ 3.50 cm

✔ Answer: 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

✔ Answer: 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 = π × r × l = π × 5 × 13 = 65π ≈ 204.20 cm²
Total Surface Area = curved + base = 65π + π×5² = 65π + 25π = 90π ≈ 282.74 cm²

✔ Answer: Curved surface area = 204.20 cm², Total surface area = 282.74 cm²

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Problem 6: Cone with r=7 cm, slant height l=25 cm → need height first? Wait — we can find volume if we have height.

But wait — they gave us slant height and radius, so we can find height using Pythagoras:

h = √(l² - r²) = √(25² - 7²) = √(625 - 49) = √576 = 24 cm

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

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

✔ Answer: Volume = 1231.50 cm³, Total surface area = 703.72 cm²

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Problem 7: Inverted 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
So Volume = 165.888π ≈ 521.15 cm³

Total Surface Area = curved + base = πrl + πr²
Need slant height l = √(r² + h²) = √(7.2² + 9.6²) = √(51.84 + 92.16) = √144 = 12 cm

So Curved SA = π×7.2×12 = 86.4π
Base SA = π×7.2² = 51.84π
Total = 86.4π + 51.84π = 138.24π ≈ 434.29 cm²

✔ Answer: Volume = 521.15 cm³, Total surface area = 434.29 cm²

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Problem 8: Composite shape — cone on top of hemisphere. Both share same radius.

Given: total height = 23 cm, cone height = 15 cm → so hemisphere height = radius = 23 - 15 = 8 cm? Wait — no!

Wait — look at diagram: it says “Cone” and “Hemisphere”, and shows a vertical line from tip to bottom labeled 23 cm, and inside the cone part, from tip to base of cone is 15 cm. So the hemisphere must be below that, and its height equals its radius.

So: radius r = ? The hemisphere’s height is equal to its radius. Since total height is 23 cm and cone height is 15 cm, then hemisphere height = 23 - 15 = 8 cm → so radius r = 8 cm.

Check: yes, because hemisphere height = radius.

So now:

Volume of cone = (1/3)πr²h_cone = (1/3)π×64×15 = (1/3)×960π = 320π
Volume of hemisphere = (2/3)πr³ = (2/3)π×512 = (1024/3)π ≈ 341.333π
Total Volume = 320π + 341.333π = 661.333π ≈ 2077.65 cm³

Total Area: This includes:
- Curved surface of cone: πrl (need slant height)
- Curved surface of hemisphere: 2πr² (not including flat base, since it’s attached to cone)
- BUT — do we include the base of the hemisphere? The diagram doesn’t show an open base — probably we assume it’s closed? Wait — in composite shapes like this, usually we don’t count internal surfaces.

Actually, looking at typical problems: when a cone sits on a hemisphere, the total surface area includes:
- Lateral (curved) surface of cone
- Curved surface of hemisphere (half sphere)
- NOT the circular base where they join (internal)
- AND NOT the flat base of the hemisphere? Wait — the diagram shows the hemisphere sitting on a surface? Actually, the label says “Total area” — likely meaning external surface only.

Standard interpretation: Total surface area = curved surface of cone + curved surface of hemisphere.

Because the flat circle between them is hidden, and the bottom of the hemisphere might be considered part of the object — but in many textbooks, for such composites, they include the hemisphere’s curved surface only, not the flat base, unless specified.

Wait — let me check the diagram again mentally: it’s a cone on top of a hemisphere, forming a sort of ice cream cone shape. Usually, you’d include:
- Cone’s lateral surface
- Hemisphere’s curved surface (the dome)
- And sometimes the flat circular base at the bottom? But in this case, since it’s sitting on something, maybe not.

But the problem says “Total area” — and in context of previous problems, for cone alone they included base, for sphere they did full surface.

Looking back at problem 5 and 6: for cone, “total surface area” included base.

Here, since it’s a composite, and the hemisphere has a flat face attached to the cone, we should NOT include that interface.

Also, the bottom of the hemisphere — is it exposed? The diagram doesn’t specify, but typically in such problems, if it’s standing on a surface, we might exclude the bottom. However, to be safe, let’s see what makes sense.

Actually, re-examining: the figure shows a vertical dimension from tip to bottom = 23 cm, and cone height = 15 cm, so hemisphere radius = 8 cm. The hemisphere is below, so its flat face is up, attached to cone, and its curved part is down. If it’s sitting on a table, the very bottom point is touching, but the entire curved surface is exposed except the attachment.

In most textbook problems, for "cone on hemisphere", total surface area = lateral surface of cone + curved surface of hemisphere. They do NOT include the flat circular base of the hemisphere because it's either internal or not part of the outer surface.

Moreover, in problem 8, it says “Total area” — and given that, I think we go with:

Total Area = πrl (cone lateral) + 2πr² (hemisphere curved)

We have r = 8 cm

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

So:

Cone lateral SA = π × 8 × 17 = 136π
Hemisphere curved SA = 2π × 8² = 2π × 64 = 128π
Total = 136π + 128π = 264π ≈ 829.38 cm²

But wait — is there a base? The hemisphere’s flat face is attached, so no. The cone’s base is also attached, so no. Only outer surfaces.

Yes.

Volume we already calculated: 320π + (2/3)π(512) = 320π + 1024/3 π = (960/3 + 1024/3)π = 1984/3 π ≈ 661.333π ≈ 2077.65 cm³

✔ Answer: Volume = 2077.65 cm³, Total area = 829.38 cm²

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Final Answers Compiled:

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 = 434.29 cm²
8) Volume = 2077.65 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 = 434.29 cm²
8) Volume = 2077.65 cm³, Total area = 829.38 cm²