Volume of Prisms and Cylinders
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.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
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?
Every prism and cylinder obeys one rule: volume equals the cross-sectional area multiplied by the perpendicular length. Unit conversions link the result to real-world capacities.
$V = Ah$ — where $A$ is the area of the uniform cross-section and $h$ is the perpendicular length. The cross-section is the shape you see when you slice perpendicular to the length.
Unit conversions: $1\text{ m}^3 = 1000\text{ L} = 1\text{ kL}$; $1\text{ L} = 1000\text{ cm}^3$; $1\text{ cm}^3 = 1\text{ mL}$.
Key facts
- $V = Ah$ applies to any prism or cylinder
- The cross-section is the shape perpendicular to the length
- Unit conversions: $1\text{ m}^3 = 1000\text{ L}$, $1\text{ L} = 1000\text{ cm}^3$
- Composite volumes are found by addition or subtraction
Concepts
- Why identifying the cross-section first simplifies every volume problem
- How a trapezoidal prism models real-world sloped containers
- Why unit consistency matters before substituting values
Skills
- Calculate volume of rectangular prisms, triangular prisms, cylinders
- Find volume of composite prisms by addition/subtraction
- Convert between cm³, m³, and litres
- Solve capacity/tank/pipe problems
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 $A$.
- Rectangular prism: cross-section is a rectangle → $A = \ell w$, so $V = \ell wh$
- Triangular prism: cross-section is a triangle → $A = \frac{1}{2}bh_{\triangle}$, then multiply by length $\ell$
- Trapezoidal prism: cross-section is a trapezium → $A = \frac{1}{2}(a+b)h$, then multiply by length
- Cylinder: cross-section is a circle → $A = \pi r^2$, then multiply by height
What to write in your book
- The universal rule: $V = Ah$ — area of cross-section × perpendicular length. Works for every prism and cylinder.
- Identify the cross-section first: slice perpendicular to the length and ask "what shape do I see?"
- Cylinder: $V = \pi r^2 h$ (circular cross-section). If given diameter, find $r = d \div 2$ first.
- Unit conversions: $1\text{ m}^3 = 1000\text{ L} = 1\text{ kL}$; $1\text{ L} = 1000\text{ cm}^3$; $1\text{ cm}^3 = 1\text{ mL}$.
Quick check: A triangular prism has cross-sectional base 6 cm, cross-sectional height 4 cm, and length 15 cm. Which expression gives its volume?
Worked examples · 3 in a row, reveal as you go
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.
A cylindrical water tank has diameter 1.4 m and height 2.2 m. Find its volume in m³ and capacity in kilolitres (kL).
What to write in your book
- Trapezoidal prism: $A = \tfrac{1}{2}(a+b)h$, then $V = A \times \text{length}$. Used for sloped containers like swimming pools.
- Cylinder: $V = \pi r^2 h$. If given diameter, $r = d \div 2$ first.
- Composite solids: split into recognisable shapes, find each volume, then add or subtract.
- Hollow cylinders (pipes): $V = \pi(R^2 - r^2)h$ where $R$ = outer radius, $r$ = inner radius.
True or false: $1\text{ m}^3$ is equal to 1000 litres.
Common errors · the 3 traps that cost marks
What to write in your book
- Always label which dimension is the cross-section height and which is the prism length.
- Check unit consistency before substituting — don't mix cm and m in the same calculation.
- $1\text{ m}^3 = 1000\text{ L}$: multiply by 1000 to go from m³ to L; divide to go from L to m³.
- Composite volumes: identify each component shape, calculate separately, then add or subtract.
Type the answer: A swimming pool (trapezoidal cross-section) has parallel sides 1.0 m and 2.5 m, horizontal length 20 m, and width 8 m. What is the volume in m³?
Quick-fire practice · 5 calculations
A brick is 22 cm × 11 cm × 7.5 cm. Find its volume in cm³.
A rectangular storage container is 2.4 m long, 1.2 m wide, and 1.5 m high. Find its volume in m³.
A can has radius 4 cm and height 12 cm. Find its volume in cm³ (to 2 d.p.).
A cylindrical fuel drum has diameter 0.6 m and height 1.0 m. Find its capacity in litres (nearest litre).
A pipe has outer radius 8 cm, inner radius 6 cm, and length 50 cm. Find the volume of metal in the pipe in cm³ (leave in terms of $\pi$).
Fill the gap: To convert a volume in m³ to litres, by . To convert cm³ to mL, the value stays the .
Earlier you predicted how to model the pool and estimate its volume. Let's confirm:
The pool's cross-section (viewed from the side) is a trapezium with parallel sides 1.2 m and 2.4 m, and horizontal length 25 m. The width is 10 m.
$A = \tfrac{1}{2}(1.2 + 2.4)(25) = \tfrac{1}{2}(3.6)(25) = 45\text{ m}^2$
$V = 45 \times 10 = 450\text{ m}^3 = \mathbf{450\,000\text{ L}}$
The trapezoidal prism model works because the depth changes uniformly — the cross-section is a consistent trapezium from side to side.
Pick your answer, then rate your confidence — that tells the system what to drill next. Each retry pulls a fresh mix from the bank.
Q1. 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. (b) Find the volume of concrete required in m³. (c) Concrete costs $220 per m³. Find the cost of the concrete. (3 marks)
Q2. 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). (b) Find the capacity in kilolitres. (3 marks)
Q3. 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. (b) Find the total cross-sectional area. (c) Find the total volume of soil required in m³. (d) Soil is sold in bags of 0.1 m³. How many bags are needed? (4 marks)
📖 Comprehensive answers (click to reveal)
Drill 1: $22 \times 11 \times 7.5 = 1815\text{ cm}^3$ · 2: $2.4 \times 1.2 \times 1.5 = 4.32\text{ m}^3$ · 3: $\pi(4)^2(12) = 192\pi \approx 603.19\text{ cm}^3$ · 4: $r = 0.3$; $V = \pi(0.09)(1.0) \approx 0.2827\text{ m}^3 = 283\text{ L}$ · 5: $\pi(64-36)(50) = 1400\pi\text{ cm}^3$
Q1 (3 marks): (a) $A = \frac{1}{2}(0.5+0.8)(1.2) = \frac{1}{2}(1.3)(1.2) = 0.78\text{ m}^2$ [1]. (b) $V = 0.78 \times 15 = 11.7\text{ m}^3$ [1]. (c) $\text{Cost} = 11.7 \times 220 = \$2574$ [1].
Q2 (3 marks): (a) $r = 1.2$; $V = \pi(1.44)(3) = 4.32\pi \approx 13.57\text{ m}^3$ [2]. (b) $13.57\text{ kL}$ (since $1\text{ m}^3 = 1\text{ kL}$) [1].
Q3 (4 marks): (a) $A_{\text{trap}} = \frac{1}{2}(1.2+2.0)(0.3) = 0.48\text{ m}^2$ [1]. (b) $A_{\text{rect}} = 1.2 \times 0.4 = 0.48$; Total $= 0.96\text{ m}^2$ [1]. (c) $V = 0.96 \times 8 = 7.68\text{ m}^3$ [1]. (d) $7.68 \div 0.1 = 76.8 \to 77\text{ bags}$ [1].
Five timed questions on prism and cylinder volumes. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering volume of prisms and cylinders questions. Pool: lessons 1–9.
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