Math worksheet titled "Volume Word Problems" featuring six questions on calculating volumes and surface areas of geometric shapes like pyramids, cones, spheres, and cylinders.

Let’s solve each problem step by step.

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Problem 1:
Calculate the volume of a pyramid with a square base of sides 7 cm and a vertical height of 11 cm.

→ Formula for volume of a pyramid:
V = (1/3) × base area × height

→ Base is square → base area = side × side = 7 × 7 = 49 cm²
→ Height = 11 cm
→ V = (1/3) × 49 × 11 = (539)/3 ≈ 179.666...
→ Rounded to 1 decimal place: 179.7 cm³

✔ Checked: 49 × 11 = 539; 539 ÷ 3 = 179.666… → correct.

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Problem 2:
Find the total surface area of a cone whose base has a radius of 2.5 cm and slant height is 10 cm. Give your answer in terms of π.

→ Total surface area of a cone = πr² + πrl
(where r = radius, l = slant height)

→ r = 2.5, l = 10
→ πr² = π × (2.5)² = π × 6.25
→ πrl = π × 2.5 × 10 = π × 25
→ Total = 6.25π + 25π = 31.25π cm²

✔ Checked: 2.5² = 6.25; 2.5×10=25; 6.25+25=31.25 → correct.

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Problem 3:
Calculate the volume of a sphere with a diameter of 40 mm.

→ First, find radius: diameter = 40 mm → radius = 20 mm
→ Volume of sphere = (4/3)πr³
→ r³ = 20³ = 8000
→ V = (4/3) × π × 8000 = (32000/3)π ≈ 10666.666...π mm³
→ But we need a numerical value? Wait — the instruction says “Leave answers to 1 decimal place.” So we must compute numerically.

→ Use π ≈ 3.1415926535
→ V = (4/3) × π × 8000 = (32000/3) × π ≈ 10666.6667 × 3.1415926535 ≈ 33510.3216 mm³
→ Rounded to 1 decimal place: 33510.3 mm³

✔ Checked: 4/3 × 8000 = 32000/3 ≈ 10666.6667; × π ≈ 33510.32 → yes.

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Problem 4:
A sphere fits exactly into a cube with sides 18 mm. Calculate the surface area of the sphere. Given your answer in squared centimetres.

→ If sphere fits exactly in cube, then diameter of sphere = side of cube = 18 mm
→ So radius = 9 mm
→ Surface area of sphere = 4πr² = 4π(9)² = 4π×81 = 324π mm²

→ Now convert to cm²:
Since 1 cm = 10 mm → 1 cm² = 100 mm²
So divide by 100: 324π / 100 = 3.24π cm²

→ Numerical value? The question doesn’t specify “in terms of π” or “decimal”, but earlier instruction says “Leave answers to 1 decimal place”. So let’s compute numerically.

→ 3.24 × π ≈ 3.24 × 3.1415926535 ≈ 10.17876
→ Rounded to 1 decimal place: 10.2 cm²

✔ Checked: 9²=81; 4×81=324; 324 mm² = 3.24 cm²? Wait — no! Mistake here!

Wait — surface area is in mm², so 324π mm². To convert to cm², since 1 cm² = 100 mm², we divide by 100 → 324π / 100 = 3.24π cm² → that’s correct.

Then 3.24 × π ≈ 10.17876 → 10.2 cm² → correct.

But wait — is it 3.24π cm² or just 10.2 cm²? The question says “calculate the surface area... give your answer in squared centimetres” — and earlier global instruction: “Leave answers to 1 decimal place.” So we should give numerical value.

✔ Final: 10.2 cm²

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Problem 5:
A metal sphere of diameter 14 cm is melted down and recast into a solid cylinder of diameter 4 cm. Work out the height of the cylinder.

→ Volume remains same when melted and recast.

→ Sphere: diameter = 14 cm → radius = 7 cm
→ Volume of sphere = (4/3)πr³ = (4/3)π(343) = (1372/3)π cm³

→ Cylinder: diameter = 4 cm → radius = 2 cm
→ Volume of cylinder = πr²h = π(4)h = 4πh

→ Set equal: 4πh = (1372/3)π
→ Divide both sides by π: 4h = 1372/3
→ h = (1372/3) ÷ 4 = 1372/(3×4) = 1372/12 = 343/3 ≈ 114.333... cm
→ Rounded to 1 decimal place: 114.3 cm

✔ Checked: 7³=343; 4/3×343=1372/3; divided by 4 → 1372/12=114.333… → correct.

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Problem 6:
What is the volume of a cone whose curved surface area is 256π cm² and slant height 32 cm?

→ Curved surface area of cone = πrl = 256π
→ r × l = 256
→ l = 32 → r = 256 / 32 = 8 cm

→ Now, to find volume, we need vertical height h.
→ Use Pythagoras: l² = r² + h²
→ 32² = 8² + h² → 1024 = 64 + h² → h² = 960 → h = √960

→ Simplify √960:
960 = 64 × 15 → √960 = 8√15

→ Volume of cone = (1/3)πr²h = (1/3)π(64)(8√15) = (1/3)π(512√15) = (512√15 / 3) π

→ But we need numerical value? And round to 1 decimal place.

→ First, √15 ≈ 3.872983
→ 512 × 3.872983 ≈ 1982.967
→ Divide by 3: ≈ 660.989
→ Multiply by π: ≈ 660.989 × 3.1415926535 ≈ 2076.5

Wait — let me recalculate carefully:

Volume = (1/3)πr²h = (1/3)π(8²)(√960) = (1/3)π(64)(√960)

√960 = √(64×15) = 8√15 ≈ 8 × 3.872983 = 30.983864

Then: 64 × 30.983864 = 1982.967296

Divide by 3: 660.9890987

Multiply by π: 660.9890987 × 3.1415926535 ≈ 2076.5 (let me compute):

660.989 × 3.1415926535:

First, 660 × 3.1415926535 ≈ 2073.451
0.989 × 3.1415926535 ≈ 3.107
Total ≈ 2076.558 → rounded to 1 decimal: 2076.6 cm³

✔ Checked: r=8, h=√(32²-8²)=√(1024-64)=√960≈30.983866
V=(1/3)π(64)(30.983866)≈ (1/3)π(1982.967)≈660.989π≈2076.56 → 2076.6 cm³

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Final Answers:

1) 179.7 cm³
2) 31.25π cm²
3) 33510.3 mm³
4) 10.2 cm²
5) 114.3 cm
6) 2076.6 cm³

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Final Answer:
1) 179.7 cm³
2) 31.25π cm²
3) 33510.3 mm³
4) 10.2 cm²
5) 114.3 cm
6) 2076.6 cm³