Mathematics Standard • Year 11 • Module 2 • Lesson 9

Volume of Prisms and Cylinders

Build fluency with the universal rule V = Ah, the cylinder formula V = πr²h, and the key capacity conversions (1 m³ = 1000 L = 1 kL; 1 L = 1000 cm³).

Build · Skill Drill

1. Quick recall

Answer each question in the space provided. 1 mark each

Q1.1 Complete each volume formula.

Any prism or cylinder (universal rule): V = ______________________

Rectangular prism (special case): V = ______________________

Cylinder (special case): V = ______________________

Q1.2 Complete each capacity conversion.

1 m³ = ____________ L = ____________ kL

1 L = ____________ cm³ 1 cm³ = ____________ mL

Q1.3 Tick the units that volume uses. (One only.)

cm cm² cm³

Stuck? Revisit lesson § Key Formulas panel and Composite & Conversions section.

2. Worked example — rectangular prism + capacity conversion

Follow each line of working. Every step has a reason on the right.

Problem. A fish tank is 80 cm long, 40 cm wide, 35 cm deep. Find the volume in cm³ and the capacity in litres.

Step 1 — Apply V = ℓwh.

V = 80 × 40 × 35

Reason: rectangular prism — identify the three dimensions.

Step 2 — Multiply pairs.

V = 3200 × 35 = 112 000 cm³

Reason: 80 × 40 = 3200, then × 35.

Step 3 — Convert to capacity (1 L = 1000 cm³).

Capacity = 112 000 ÷ 1000 = 112 L

Reason: divide by 1000 to convert cm³ to litres.

Conclusion. V = 112 000 cm³, capacity = 112 L.

3. Faded example — cylinder + capacity

A cylindrical water tank has diameter 1.4 m and height 2.2 m. Find the volume in m³ (to 2 d.p.) and the capacity in kilolitres. Fill in each blank line. 4 marks

Step 1 — Find the radius (always halve d).

r = ____ ÷ 2 = ____ m

Step 2 — Apply V = πr²h: V = π × ( ____ )² × ____ = ____ π m³

Step 3 — Evaluate to 2 d.p.: V ≈ ____________ m³

Step 4 — Convert to kL (1 m³ = 1 kL): Capacity = ____________ kL

Conclusion. V ≈ ________ m³, capacity ≈ ________ kL.

Stuck? Revisit lesson § Worked Example 3 — Cylinder. 1 m³ = 1 kL so the numerical value doesn't change between the two units.

4. Graduated practice — prisms, cylinders and conversions

Show your working. Use consistent units before substituting.

Foundation — single-step volumes (4 questions)

Q	Problem	Answer
4.1 1	A brick is 22 cm × 11 cm × 7.5 cm. Find its volume in cm³.
4.2 1	A rectangular container: 2.4 m × 1.2 m × 1.5 m. Find its volume in m³.
4.3 1	A can has r = 4 cm and h = 12 cm. Find its volume in exact form (terms of π).
4.4 1	A triangular prism has cross-sectional area A = 20 cm² and length 20 cm. Find its volume.

Standard — typical HSC difficulty (6 questions)

Find the cross-section area first when the cross-section is not a rectangle.

4.5 A triangular prism has a right-angled cross-section with legs 5 cm and 8 cm. The prism is 20 cm long. Find V. 2 marks

4.6 A concrete ramp has trapezoidal cross-section with parallel depths 0.3 m and 0.8 m, width 4 m, and length 6 m. Find V in m³. 3 marks

4.7 A cylindrical fuel drum has diameter 0.6 m and height 1.0 m. Find capacity in litres, to the nearest litre. 3 marks

4.8 A rectangular tank holds 6000 L. Its base is 2.5 m × 1.2 m. Find its height in metres. 2 marks

4.9 Convert 0.0025 m³ to litres. 1 mark

4.10 Convert 85 000 cm³ to litres. 1 mark

Extension — reverse and composite (2 questions)

4.11 A pipe (hollow cylinder) has outer radius 8 cm, inner radius 6 cm, and length 50 cm. Find the volume of metal (in terms of π and to 2 d.p.). 3 marks

4.12 A swimming pool has trapezoidal cross-section with parallel sides 1.2 m (shallow end) and 2.0 m (deep end), horizontal length 25 m. The pool is 10 m wide. Find the capacity in kL. 3 marks

Stuck on 4.11? Volume of pipe metal = π(R² − r²) × length. (Annulus cross-section × length.)

5. Self-check the easy 3

Tick the first three once you've checked your method works.

For 4.1 (brick) my units are cm³ (cubic), not cm or cm².

For 4.3 (can) I substituted r = 4 (not d = 8) into πr²h and left my answer as 192π cm³.

For 4.10 (85 000 cm³) I divided by 1000 (not by 100 or 100 000) to get 85 L.

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Got it Partly Lost

What I'll revisit before next class:

Answers — Do not peek before attempting

Q1.1 — Three formulas

Universal rule: V = Ah. Rectangular prism: V = ℓwh. Cylinder: V = πr²h.

Q1.2 — Capacity conversions

1 m³ = 1000 L = 1 kL. 1 L = 1000 cm³. 1 cm³ = 1 mL.

Q1.3 — Volume units

cm³ (cubic). cm² is for area, cm is for length.

Q3 — Faded example (cylinder d = 1.4 m, h = 2.2 m)

Step 1: r = 1.4 ÷ 2 = 0.7 m.
Step 2: V = π(0.7)²(2.2) = π(0.49)(2.2) = 1.078π m³.
Step 3: V ≈ 3.39 m³ (2 d.p.).
Step 4: Capacity ≈ 3.39 kL.

Q4.1 — Brick 22 × 11 × 7.5

V = 22 × 11 × 7.5 = 1815 cm³.

Q4.2 — Container 2.4 × 1.2 × 1.5 m

V = 2.4 × 1.2 × 1.5 = 4.32 m³.

Q4.3 — Can r = 4, h = 12

V = π(16)(12) = 192π cm³ (≈ 603.19 cm³).

Q4.4 — Prism A = 20, L = 20

V = A × L = 20 × 20 = 400 cm³.

Q4.5 — Right-triangular prism (5, 8, L = 20)

A = ½(5)(8) = 20 cm². V = 20 × 20 = 400 cm³.

Q4.6 — Concrete ramp (trapezoidal)

A = ½(0.3 + 0.8)(4) = ½(1.1)(4) = 2.2 m². V = 2.2 × 6 = 13.2 m³.

Q4.7 — Fuel drum d = 0.6 m, h = 1.0 m

r = 0.3 m. V = π(0.09)(1.0) = 0.09π ≈ 0.2827 m³ = 282.74 L ≈ 283 L.

Q4.8 — Tank holds 6000 L

6000 L = 6 m³. Base area = 2.5 × 1.2 = 3 m². h = 6 ÷ 3 = 2 m.

Q4.9 — 0.0025 m³ to L

0.0025 × 1000 = 2.5 L.

Q4.10 — 85 000 cm³ to L

85 000 ÷ 1000 = 85 L.

Q4.11 — Pipe metal (R = 8, r = 6, L = 50)

Cross-section (annulus) = π(64 − 36) = 28π cm². V = 28π × 50 = 1400π cm³ ≈ 4398.23 cm³.

Q4.12 — Swimming pool (trapezoidal cross-section)

Cross-section area = ½(1.2 + 2.0)(25) = ½(3.2)(25) = 40 m². V = 40 × 10 = 400 m³ = 400 kL.