← Unit 4 Volume of Prisms and Cylinders

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Volume of Prisms and Cylinders

⏱ 25 min📚 Year 9📈
Think First

A water tank is a cylinder with radius 2 m and height 3 m. How many litres does it hold? (1 m³ = 1000 L)

💡 Revision: Ensure you know area formulas and understand that volume = area × height.

Learning Intentions

Know

  • Volume formula for prisms
  • Volume formula for cylinders
  • Composite solids
  • Unit conversions

Understand

  • Why volume scales with the cube of linear dimensions
  • How to decompose complex shapes

Can Do

  • Calculate volume of any prism
  • Calculate volume of cylinders
  • Find volume of composite solids
  • Convert between cm³ and litres
CompositeDecomposeCapacityLitres
Learn Phase
1

Volume of Prisms

Cross-section × height

For any prism:

Volume of Prism

$V = A_{ ext{cross-section}} imes h$

This works because you are stacking identical cross-sections.

Rectangular prism: $V = lwh$

Triangular prism: $V = rac{1}{2}bh imes ext{length}$

2

Volume of Cylinders

Circular cross-section

For a cylinder:

Volume of Cylinder

$V = pi r^2 h$

The $pi r^2$ is the area of the circular base.

Example: Radius 5 cm, height 10 cm:

$V = pi imes 25 imes 10 = 250pi approx 785.4$ cm³

3

Composite Solids

Break into simpler parts

A composite solid is made of two or more simple solids.

To find volume:

  1. Identify the component shapes
  2. Calculate each volume separately
  3. Add (or subtract) as appropriate

Example: A house-shaped solid = rectangular prism + triangular prism.

✓

Check Understanding

Try it yourself

A cylindrical tank has radius 1.5 m and height 2 m. Find its capacity in litres.

Worked Example

Volume Calculations

1

A rectangular prism is 8 cm × 5 cm × 4 cm. Find its volume.

$V = 8 imes 5 imes 4 = 160$ cm³

2

A cylinder has radius 3 cm and height 7 cm. Find its volume.

$V = pi imes 3^2 imes 7 = 63pi approx 197.9$ cm³

3

A composite solid: rectangular base 6×4×3 with a cylinder (r=2, h=3) on top. Find total volume.

Rectangular: $6 imes 4 imes 3 = 72$ cm³

Cylinder: $pi imes 4 imes 3 = 12pi approx 37.7$ cm³

Total: $72 + 37.7 = 109.7$ cm³

Common Misconceptions

✕

Use diameter instead of radius in $pi r^2$. Always check: is the given measurement radius or diameter?

✕

Forget to convert units. 1 m³ = 1,000,000 cm³ = 1000 L. Always check required units.

✕

Add volumes when you should subtract. If a hole is drilled through a solid, subtract the hole's volume.

Your Turn

Practice — Volume Practice

Work through each question in your book or digitally. Answers are in the Questions phase.

1Find the volume of a triangular prism with base 6 cm, height 4 cm, length 10 cm.
2A cylindrical water tank (r=2 m, h=5 m) is half full. How many litres?
3A solid is a cube (side 5 cm) with a cylindrical hole (r=1 cm, h=5 cm) through it. Find remaining volume.
Real-World Anchor

Engineering and Design

Civil engineers calculate volumes for concrete pours, reservoir capacities, and earthworks. The Snowy Hydro scheme required precise volume calculations for tunnels and dams across the Australian Alps.

📓 Copy Into Your Books

▼

Prism

  • $V = A_{ ext{cross}} imes h$
  • Units: cm³, m³

Cylinder

  • $V = pi r^2 h$
  • $SA = 2pi r^2 + 2pi r h$

Composite

  • Break into parts
  • Add or subtract volumes
  • Watch units
Questions Phase
Check Your Understanding
Answer all questions correctly to unlock the Game phase.
Volume of rectangular prism 3×4×5 cm:
Cylinder radius 2 cm, height 5 cm. Volume:
1 m³ equals:
Triangular prism: base 8, height 3, length 5. Volume:
A sphere is NOT a prism because:
A tank (cylinder, r=1 m, h=2 m) holds:
Cube side doubled. Volume becomes:
Composite solid: subtract volume when:
1Find the volume of a cylinder with radius 4 cm and height 10 cm. Give answer in terms of $pi$.
2A swimming pool is 10 m × 5 m × 2 m. How many litres of water does it hold?
3Explain why doubling all dimensions of a cube increases its volume 8-fold.

Comprehensive Answers

1Cylinder radius 4, height 10. V?
$V = pi(4)^2(10) = 160pi$ cm³.
2Pool 10×5×2. Litres?
$V = 100$ m³ = $100 imes 1000 = 100,000$ L.
3Double cube dimensions.
Volume scales as $(2)^3 = 8$. New volume = 8× original.
MC 1Prism 3×4×5.
$60$ cm³. Answer: C
MC 2Cylinder volume.
$pi(4)(5) = 20pi$. Answer: B
MC 31 m³ = ? L.
$1000$ L. Answer: B
MC 4Triangular prism.
$ rac{1}{2}(8)(3) imes 5 = 60$. Answer: B
MC 5Sphere not prism.
Cross-section changes. Answer: B
MC 6Tank capacity.
$V = pi(1)^2(2) = 2pi$ m³ = $2000pi$ L. Answer: A
MC 7Cube doubled.
$2^3 = 8$. Answer: D
MC 8Subtract volume when.
Removing a hole. Answer: B
SA 1Cylinder V radius 4, height 10.
$160pi$ cm³.
SA 2Pool litres.
$100,000$ L.
SA 3Double cube dimensions.
Volume scales by $2^3 = 8$.
Game Phase
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