Rectangular prism
V=lwh
SA=2(lw+lh+wh)
Learning goals: what you will learn
- explain the central idea: Solve surface-area and volume problems for composite objects by decomposing solids, managing joined faces and using appropriate units.
- choose and apply an appropriate method without relying on keyword matching
- check results using units, substitution, estimation, a second representation or contextual reasonableness
- justify a conclusion and communicate limitations where the context requires them
Prerequisite knowledge
Revise areas of rectangles and circles, volume of prisms and cylinders, metric conversions, π, and the difference between square and cubic units.
Concept teaching
For volume, decompose the object into solids and add/subtract their volumes. For surface area, count only the external faces; joined faces inside a composite object are not exposed and must not be counted.
Prism
V=area of cross-section × length.
Cylinder
V=πr²h
Curved area=2πrh
Total SA=2πr²+2πrh
Composite strategy
Sketch → label → decompose → convert units → calculate pieces → remove internal faces → state cm²/m² or cm³/m³.
Worked examples
1. Add two prism volumes
A shape is made from 8×4×3 cm and 5×4×2 cm prisms.
V=8×4×3+5×4×2=96+40=136 cm³.
2. Subtract a cut-out
A 10×8×6 cm block has a 4×3×6 cm rectangular tunnel removed.
V=480−72=408 cm³.
3. Prism plus cylinder
A 10×6×4 cm rectangular base supports a cylinder r=2 cm, h=5 cm.
V=240+20π≈302.83 cm³.
4. Joined-face surface area
Two cubes of side 3 cm joined face-to-face have 10 exposed square faces, not 12.
SA=10×9=90 cm².
5. Cylinder on a prism
When a cylinder sits on a prism, the circular contact face is internal. Add the cylinder's curved area and top circle, not both circular ends.
6. Unit conversion
A tank volume of 2.4 m³ equals 2400 L because 1 m³=1000 L.
7. Estimate a real object
Model a thermos as a cylinder r=4 cm, h=25 cm. Approximate capacity V=π(4²)(25)=400π≈1257 cm³≈1.26 L. State that walls/shape details are ignored.
8. Roof rainfall
A 120 m² roof receives 18 mm rain. Convert 18 mm=0.018 m.
Water volume≈120×0.018=2.16 m³=2160 L before losses.
9. Storage constraint
A cylindrical tank must fit a 1.8 m diameter space, so r≤0.9 m. With height 2 m, maximum ideal volume is π(0.9²)(2)≈5.09 m³.
10. Freezer-space decision
If 180 tubs each occupy 0.0045 m³, required volume=0.81 m³. A 0.75 m³ freezer is insufficient by 0.06 m³; extra space is needed.
Common misconceptions and corrections
- Adding areas when the question asks for volume, or vice versa.
- Counting internal joined faces in surface area.
- Mixing cm and m before calculating.
- Forgetting squared/cubed units.
- Using a real-world model without stating simplifications or wasted space.
Guided practice
A solid is a 10 cm × 6 cm × 4 cm prism with a 2 cm-radius, 5 cm-high cylinder attached on top. First find total volume. Then list exactly which surfaces are external before attempting surface area.
Teacher check: require a written method choice and one verification step before revealing the worked solution.
Independent practice
- Find the volume of a 9×5×4 cm prism.
- Subtract a 3×2×5 cm cut-out from a 10×6×5 cm block.
- Find the volume of a cylinder r=3 cm, h=8 cm.
- Two 4 cm cubes are joined face-to-face. Find surface area.
- Convert 1.75 m³ to litres.
- A roof area is 95 m² and rainfall is 12 mm. Estimate litres collected before losses.
- Estimate capacity of a cylinder r=5 cm, h=20 cm in litres.
- Explain why a contact face is removed from external surface area.
- A tank r=1.2 m, h=1.5 m: find volume to 2 d.p.
- Name one assumption in modelling a freezer or rainwater tank.
Check answers and explanations
- 180 cm³.
- 300−30=270 cm³.
- 72π≈226.19 cm³.
- 10 exposed faces×16=160 cm².
- 1750 L.
- 95×0.012=1.14 m³=1140 L.
- 500π cm³≈1571 cm³=1.57 L.
- It lies inside the composite object and is not exposed.
- π(1.2²)(1.5)≈6.79 m³.
- Example: ideal packing/no wasted space or all roof runoff is captured.
Reasoning and problem-solving task
A community event needs storage for 240 containers, each modelled as a 12 cm × 8 cm × 6 cm rectangular prism. Compare the theoretical packed volume with a freezer offering 1.6 m³ usable space. Then explain why volume alone may not guarantee the containers fit.
Important questions and answers
- Find the volume of a 9×5×4 cm prism.
- 180 cm³.
- Subtract a 3×2×5 cm cut-out from a 10×6×5 cm block.
- 300−30=270 cm³.
- Find the volume of a cylinder r=3 cm, h=8 cm.
- 72π≈226.19 cm³.
Assessment-style questions
A rainwater tank is modelled as a rectangular prism 1.8 m × 1.2 m × 0.7 m with a half-cylinder roof of radius 0.6 m and length 1.8 m. Develop an expression for total capacity, calculate it and state two modelling assumptions. [6 marks]
Marking focus: method selection, mathematically correct working, interpretation and justification.
Review hints
- draw and label before calculating
- for surface area, mark exposed faces only
- convert all lengths to one unit first
- keep π until the final rounding step
Exit ticket: mastery check
- I can explain the concept without copying a formula sheet.
- I can solve a routine example and check the result.
- I can choose a method in an unfamiliar problem.
- I can explain a common error and correct it.
- I can connect the answer back to the context, including units or limitations.
Teacher and parent guidance
For teachers
Use physical boxes or quick sketches to make joined faces visible. Ask students to annotate every exposed face before writing any surface-area formula; this prevents the most common error.
For parents and carers
Ask your child to point to the parts they are adding or subtracting and to explain whether the answer should be in square or cubic units before they calculate.
Curriculum alignment
Australian Curriculum: AC9M10M01 — Year 10 Measurement. Solve surface-area and volume problems for composite objects by decomposing solids, managing joined faces and using appropriate units.
Victoria: VC2M10M01 — Level 10 Measurement
NSW: Stage 5 Core — Length, area and volume; Path — Further area and volume
Alignment explanation: The explicit teaching and worked examples address the Australian Curriculum concept directly. The Victorian mapping follows the current Version 2.0 descriptor structure; where Victoria combines or extends content, that difference is stated rather than hidden. NSW uses a Stage 5 Core–Paths structure, so this page maps to the relevant content group(s) and Working mathematically processes instead of inventing a Year 10 one-to-one code.
| Lesson component | Australian Curriculum | Victoria | NSW |
|---|---|---|---|
| Explicit concept teaching and worked examples | AC9M10M01 | VC2M10M01 — Level 10 Measurement | Stage 5 Core — Length, area and volume; Path — Further area and volume |
| Guided and independent practice | Builds fluency and application for AC9M10M01 | Practises the mapped Level 10/10A knowledge as applicable | Practises the mapped Stage 5 Core/Path content |
| Reasoning and assessment tasks | Applies reasoning/problem solving in the descriptor context | Supports Victorian reasoning and modelling expectations | Embeds Working mathematically: reasoning, problem solving and communication |
Australian Curriculum elaborations
AC9M10M01: solve problems involving the surface area and volume of composite objects using appropriate units.
- E1: determine surface area and volume of composite solids made from right prisms and cylinders by considering component solids. Examples 1–5.
- E2: estimate surface area and volume of composite objects in practical contexts. Example 7.
- E3: use mathematical modelling for surface-area/volume decisions such as rooftop rainwater storage or freezer capacity. Examples 8–10.
Practice and teaching resources
Official curriculum references
- Australian Curriculum Version 9 — ACARA
- Victorian Curriculum Mathematics Version 2.0 — VCAA
- NSW Mathematics K–10 Syllabus (2022) — NSW Curriculum/NESA
Official wording is paraphrased on SkillrHub; use the linked curriculum sites as the source of record.
Other curriculum comparisons retained
Closest alignment: NSW Stage 5 and Victorian Level 10 Measurement; US CCSS HSG-GMD.A/B and modelling applications; UK GCSE mensuration; comparable Canadian/NZ/CBSE surface-area and volume work.
🎥 Optional Video Lesson
The SkillrHub lesson remains the primary learning resource. This optional video reinforces the explanation; you can complete the lesson and practice without watching.
Before you watch:
- Pause after each worked example.
- Try the examples yourself.
- Return to the SkillrHub lesson before continuing.
Recommended: Volumes of Composite Solids
Eddie Woo — Splitting a composite solid into familiar solids before calculating volume.
As you watch: Which dimensions belong to each component, and which volumes must be added or subtracted?
Load video player Loads YouTube in this lesson. See the video notice below.
Try it: Sketch a cylinder joined to a hemisphere, label dimensions and write a volume calculation with consistent units.
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Curriculum equivalents: Victoria, NSW and international
Curriculum equivalents for Solve problems involving the surface area and volume of composite...
Mapped skill: solve problems involving the surface area and volume of composite objects using appropriate units
These references identify matching or closely related learning. Curriculum sequence, terminology and depth vary, so teachers should use the mapped skill and lesson difficulty to confirm suitability.
| Region | Curriculum framework | Closest level or code |
|---|---|---|
| Australia | Australian Curriculum v9.0 | AC9M10M01 · Year 10 |
| Victoria | Victorian Curriculum F–10 Version 2.0 — Mathematics | VC2M10M01 · Level 10 |
| New South Wales | NSW Mathematics K–10 Syllabus (2022) | MA5-VOL-C-01 + MA5-ARE-C-01 · Stage 5 |
| United States (USA) | Common Core State Standards for Mathematics | Grades 9–10 band |
| Canada (Ontario) | Ontario Curriculum — Mathematics | Grade 10 |
| United Kingdom (England) | National Curriculum in England — Mathematics | Year 11, Key Stage 4 |
| India | NCERT / CBSE — Mathematics | Class 10 |
Australian Curriculum v9.0 is the canonical source for this SkillrHub lesson. Victoria and NSW entries name the closest published state codes or outcomes; international entries are planning references rather than claims of identical curricula.
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Topic reference: AC9M10M01 — Composite Surface Area and Volume — AC9M10M01
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