Volume = cross-sectional area × length. Identify the uniform cross-section first — everything else follows. Unit conversions between cm³, m³, and litres are critical for practical problems.
Use the PDF for classwork, homework or revision. It includes key ideas, activities, questions, an extend task and success-criteria proof.
A swimming pool is 25 m long, 10 m wide, and has a depth that slopes from 1.2 m at the shallow end to 2.4 m at the deep end. How would you estimate how many litres of water it holds? What shape would you use, and why?
Type your initial response below — you will revisit this at the end of the lesson.
Write your initial response in your book. You will revisit it at the end of the lesson.
Come back to this at the end of the lesson.
Wrong: A percentage over 100% is impossible.
Right: Percentages over 100% are valid and common in contexts like percentage increase, profit margins, and scale factors.
Prism Volume — $V = Ah$
Always check your units before substituting into formulas. Converting to consistent units is a common source of errors in assessment tasks.
Every prism — regardless of its cross-sectional shape — has volume equal to the cross-sectional area multiplied by the perpendicular length.
The key is correctly identifying the cross-section. Ask: "If I sliced this solid perpendicular to its longest dimension, what shape would I see?" That area is $A$.
A fish tank is 80 cm long, 40 cm wide, and 35 cm deep. Find its volume in cm³ and capacity in litres.
A triangular prism has a right-triangular cross-section with base 6 cm and perpendicular height 4 cm. The prism is 15 cm long. Find its volume.
Cylinders — $V = \pi r^2 h$
A cylinder is just a circular prism. Its cross-section is always a circle, so $A = \pi r^2$ and the volume formula becomes $V = \pi r^2 h$.
A cylindrical water tank has diameter 1.4 m and height 2.2 m. Find its volume in m³ and capacity in kilolitres (kL).
Composite Volumes and Unit Conversions
A composite solid is built from two or more recognisable shapes. Identify each component, calculate its volume separately, then add (or subtract if one is removed/hollow).
A swimming pool is 20 m long and 8 m wide. The depth slopes uniformly from 1.0 m at the shallow end to 2.5 m at the deep end. Find the volume in m³ and capacity in kL.
Section A — Rectangular Prisms
Section B — Triangular and Trapezoidal Prisms
Section C — Cylinders and Conversions
$22 \times 11 \times 7.5 = \mathbf{1815 \text{ cm}^3}$
$2.4 \times 1.2 \times 1.5 = \mathbf{4.32 \text{ m}^3}$
$6000 \text{ L} = 6 \text{ m}^3$; $h = 6 \div (2.5 \times 1.2) = 6 \div 3 = \mathbf{2 \text{ m}}$
$A = \frac{1}{2}(5)(8) = 20$; $V = 20 \times 20 = \mathbf{400 \text{ cm}^3}$
$A = \frac{1}{2}(0.3+0.8)(4) = \frac{1}{2}(1.1)(4) = 2.2$; $V = 2.2 \times 6 = \mathbf{13.2 \text{ m}^3}$
$A = \frac{1}{2}(1.2+2.0)(25) = \frac{1}{2}(3.2)(25) = 40$; $V = 40 \times 10 = 400 \text{ m}^3 = \mathbf{400 \text{ kL}}$
$V = \pi(4)^2(12) = 192\pi \approx \mathbf{603.19 \text{ cm}^3}$
$r = 0.3$; $V = \pi(0.09)(1.0) = 0.09\pi \approx 0.2827 \text{ m}^3 = \mathbf{283 \text{ L}}$
$V = \pi(8^2 - 6^2)(50) = \pi(64-36)(50) = \pi(28)(50) = \mathbf{1400\pi \text{ cm}^3}$
$0.0025 \times 1000 = \mathbf{2.5 \text{ L}}$
$85\,000 \div 1000 = \mathbf{85 \text{ L}}$
$4.5 \text{ kL} = 4.5 \text{ m}^3 = 4.5 \times 10^6 \text{ cm}^3 = \mathbf{4\,500\,000 \text{ cm}^3}$
Look back at what you wrote in the Think First section. What has changed? What did you get right? What surprised you?
Multiple Choice
1 A triangular prism has a cross-section that is a right-angled triangle with legs 3 m and 4 m. The prism is 10 m long. Its volume is:
? Regarding this topic, 1 A triangular prism has a cross-section that is a right-angled triangle with legs 3 m and 4 m. The prism is 10 m long. Its volume is:
2 A cylinder has diameter 1.2 m and height 0.8 m. Its volume in litres (to the nearest litre) is:
? Regarding this topic, 2 A cylinder has diameter 1.2 m and height 0.8 m. Its volume in litres (to the nearest litre) is:
3 A trough has a trapezoidal cross-section with parallel sides 0.4 m and 0.6 m, and a perpendicular height of 0.3 m. The trough is 2 m long. Its capacity in litres is:
? Regarding this topic, 3 A trough has a trapezoidal cross-section with parallel sides 0.4 m and 0.6 m, and a perpendicular height of 0.3 m. The trough is 2 m long. Its capacity in litres is:
Short Answer
SA 4 3 marks A concrete retaining wall has a trapezoidal cross-section with parallel sides of 0.5 m and 0.8 m, a perpendicular height of 1.2 m, and a length of 15 m.
(a) Find the area of the trapezoidal cross-section. (1 mark)
(b) Find the volume of concrete required in m³. (1 mark)
(c) Concrete costs $\$220$ per m³. Find the cost of the concrete. (1 mark)
$A = \frac{1}{2}(0.5+0.8)(1.2) = \frac{1}{2}(1.3)(1.2) = \mathbf{0.78 \text{ m}^2}$
$V = 0.78 \times 15 = \mathbf{11.7 \text{ m}^3}$
$\text{Cost} = 11.7 \times 220 = \mathbf{\$2574}$
SA 5 3 marks A cylindrical rainwater tank has an internal diameter of 2.4 m and a height of 3.0 m.
(a) Find the volume of the tank in m³ (correct to 2 decimal places). (2 marks)
(b) Find the capacity in kilolitres. (1 mark)
$r = 1.2$; $V = \pi(1.2)^2(3.0) = \pi(1.44)(3) = 4.32\pi \approx \mathbf{13.57 \text{ m}^3}$
$\mathbf{13.57 \text{ kL}}$ (since $1 \text{ m}^3 = 1 \text{ kL}$)
SA 6 4 marks A garden bed has a composite cross-section consisting of a rectangle (1.2 m wide, 0.4 m deep) sitting on top of a trapezium (parallel sides 1.2 m and 2.0 m, height 0.3 m). The garden bed is 8 m long.
(a) Find the area of the trapezoidal portion. (1 mark)
(b) Find the total cross-sectional area. (1 mark)
(c) Find the total volume of soil required in m³. (1 mark)
(d) Soil is sold in bags of 0.1 m³. How many bags are needed? (1 mark)
$A_{\text{trap}} = \frac{1}{2}(1.2+2.0)(0.3) = \frac{1}{2}(3.2)(0.3) = \mathbf{0.48 \text{ m}^2}$
$A_{\text{rect}} = 1.2 \times 0.4 = 0.48$; Total $= 0.48 + 0.48 = \mathbf{0.96 \text{ m}^2}$
$V = 0.96 \times 8 = \mathbf{7.68 \text{ m}^3}$
$7.68 \div 0.1 = 76.8$ → round up → $\mathbf{77 \text{ bags}}$
Volume of Prisms and Cylinders