Australian Curriculum v9 / ACiQ Year 10 Mathematics - Unit 2 - Composite surface area and volume

Composite surface area and volume

Solve surface-area and volume problems involving composite solids and state assumptions used in models.

Updated 2026-07-26 - 12 min read

Composite surface area and volume is taught here as a connected set of decisions, not a list of facts. Work through the prerequisite recall, explicit models, carefully faded examples, misconception repairs and transfer task before using the target in Check, Practice, Review or Rapid Revision.

This note is designed to work with the guided lessons, curated practice, flashcards, Tutor context, Review and Rapid Revision for the same canonical target. The same three evidence checks are used throughout, so feedback can route a learner back to the precise idea that needs repair.

Calculate composite-solid volume

Decompose a solid into non-overlapping familiar solids and add or subtract their volumes exactly once. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate composite-solid volume and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The decomposition must partition the model; overlaps are removed or counted once.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 1.1

A solid combines cuboids 4×3×2 cm and 2×2×5 cm without overlap. Find volume.

Step 1 - identify the governing idea: Decompose a solid into non-overlapping familiar solids and add or subtract their volumes exactly once.

Step 2 - apply it to this evidence: The volumes are 24 cm³ and 20 cm³.

Result: 44 cm³

The relationship is visible in the working: The volumes are 24 cm³ and 20 cm³. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 22 cm³ — It does not agree with the required relationship: The volumes are 24 cm³ and 20 cm³.
  • 88 cm³ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 34 cm³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.2

A cylinder of radius 3 cm and height 8 cm has a cylindrical hole radius 1 cm through it. Find exact volume.

Step 1 - identify the governing idea: Decompose a solid into non-overlapping familiar solids and add or subtract their volumes exactly once.

Step 2 - apply it to this evidence: Subtract π(1²)(8) from π(3²)(8): 72π − 8π.

Result: 64π cm³

The relationship is visible in the working: Subtract π(1²)(8) from π(3²)(8): 72π − 8π. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 80π cm³ — It does not agree with the required relationship: Subtract π(1²)(8) from π(3²)(8): 72π − 8π.
  • 16π cm³ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 72π cm³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 1.3

A 10 cm cube has a 4 cm cube removed. Find remaining volume.

Step 1 - identify the governing idea: Decompose a solid into non-overlapping familiar solids and add or subtract their volumes exactly once.

Step 2 - apply it to this evidence: 1000 − 64 = 936 cm³.

Result: 936 cm³

The relationship is visible in the working: 1000 − 64 = 936 cm³. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 960 cm³ — It does not agree with the required relationship: 1000 − 64 = 936 cm³.
  • 1064 cm³ — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 600 cm³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Calculate exposed surface area

Sum only exposed faces; shared joins and internal cut faces are excluded unless the context exposes them. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by calculate exposed surface area and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: Faces at a join become internal and must be removed from the exposed total.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 2.1

Two 3 cm cubes are joined face-to-face. Find exposed surface area.

Step 1 - identify the governing idea: Sum only exposed faces; shared joins and internal cut faces are excluded unless the context exposes them.

Step 2 - apply it to this evidence: Two cubes total 108 cm²; the two joined 9 cm² faces are removed.

Result: 90 cm²

The relationship is visible in the working: Two cubes total 108 cm²; the two joined 9 cm² faces are removed. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 108 cm² — It does not agree with the required relationship: Two cubes total 108 cm²; the two joined 9 cm² faces are removed.
  • 99 cm² — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 54 cm² — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.2

A closed cylinder has r=2 cm and h=5 cm. Find exact surface area.

Step 1 - identify the governing idea: Sum only exposed faces; shared joins and internal cut faces are excluded unless the context exposes them.

Step 2 - apply it to this evidence: 2πr² + 2πrh = 8π + 20π.

Result: 28π cm²

The relationship is visible in the working: 2πr² + 2πrh = 8π + 20π. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 20π cm² — It does not agree with the required relationship: 2πr² + 2πrh = 8π + 20π.
  • 24π cm² — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 40π cm² — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 2.3

An open-top box is 6×4×3 cm. Find surface area.

Step 1 - identify the governing idea: Sum only exposed faces; shared joins and internal cut faces are excluded unless the context exposes them.

Step 2 - apply it to this evidence: Base 24 plus sides 36 and 24 gives 84; the top is omitted.

Result: 84 cm²

The relationship is visible in the working: Base 24 plus sides 36 and 24 gives 84; the top is omitted. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 108 cm² — It does not agree with the required relationship: Base 24 plus sides 36 and 24 gives 84; the top is omitted.
  • 72 cm² — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 144 cm² — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Model practical composite solids

Translate the object into idealised solids, use consistent internal or external dimensions and state omissions such as wall thickness. This relationship must be selected from the quantities and conditions in the problem, then checked against the context.

A dependable reasoning routine

  1. Name the unknowns, units and constraints before calculating.
  2. Choose the relationship represented by model practical composite solids and state why it applies.
  3. Keep exact values for as long as possible, show substitutions and preserve units through each step.
  4. Check the result by substitution, estimation, an alternative representation or the original context.

Repair: The calculation may be exact for the mathematical model while the model itself remains an approximation.

The repair matters because the shortcut may appear to work in one familiar example while failing when the method, representation, scale, constraint or accuracy requirement changes. Use the routine above to make the reasoning visible enough for another learner to verify.

Example 3.1

A tank's capacity is required and wall thickness is known. Which dimensions should be used?

Step 1 - identify the governing idea: Translate the object into idealised solids, use consistent internal or external dimensions and state omissions such as wall thickness.

Step 2 - apply it to this evidence: Capacity measures usable internal space.

Result: Internal dimensions

The relationship is visible in the working: Capacity measures usable internal space. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • External dimensions — It does not agree with the required relationship: Capacity measures usable internal space.
  • Average any dimensions — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Only height — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.2

A dome is modelled as a hemisphere. What limitation should be stated?

Step 1 - identify the governing idea: Translate the object into idealised solids, use consistent internal or external dimensions and state omissions such as wall thickness.

Step 2 - apply it to this evidence: Geometric idealisation introduces model error.

Result: The real dome may not be a perfect hemisphere

The relationship is visible in the working: Geometric idealisation introduces model error. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • π is inaccurate — It does not agree with the required relationship: Geometric idealisation introduces model error.
  • Hemisphere formulas have no units — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • Volume cannot be estimated — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Example 3.3

A composite model gives 2.846 m³ from dimensions measured to 1 cm. How should it be reported?

Step 1 - identify the governing idea: Translate the object into idealised solids, use consistent internal or external dimensions and state omissions such as wall thickness.

Step 2 - apply it to this evidence: The reporting precision should reflect the input measurements.

Result: About 2.85 m³

The relationship is visible in the working: The reporting precision should reflect the input measurements. Check the units and substitute or estimate where possible. A plausible-looking number is not enough unless it satisfies the original conditions.

Why the alternatives fail:

  • 2.846000 m³ exactly — It does not agree with the required relationship: The reporting precision should reflect the input measurements.
  • 3 m³ exactly — It changes or misses a condition in the question. Reapply the relationship and retain the stated units or accuracy.
  • 2846 m³ — A substitution, estimate, ordering or unit check rejects this result; it does not reproduce the conditions in the prompt.

Retrieval check

Try these without looking back at the examples.

  1. A cylinder of radius 3 cm and height 8 cm has a cylindrical hole radius 1 cm through it. Find exact volume.
  2. A closed cylinder has r=2 cm and h=5 cm. Find exact surface area.
  3. A dome is modelled as a hemisphere. What limitation should be stated?

Answers

  1. 64π cm³ — Subtract π(1²)(8) from π(3²)(8): 72π − 8π.
  2. 28π cm² — 2πr² + 2πrh = 8π + 20π.
  3. The real dome may not be a perfect hemisphere — Geometric idealisation introduces model error.

Transfer task

Find an unfamiliar example from school, daily life, a credible news source or another subject. Explain which of the three evidence checks applies. Complete the task, then audit your own response: identify the evidence used, the relationship applied, one plausible misconception and the final reasonableness check. If a peer could not reproduce your reasoning, add the missing step.

Sources