Australian Curriculum v9 / ACiQ Year 7 Mathematics - Unit 0 - Volumes of rectangular and triangular prisms

Volumes of rectangular and triangular prisms

Calculate volumes of right rectangular and triangular prisms and solve practical volume problems.

Updated 2026-07-24 - 6 min read

Calculate volumes of right rectangular and triangular prisms and solve practical volume problems. This draft teaching resource is aligned to the cited Year 7 content descriptions and still requires named human curriculum approval.

Name what each length, angle, coordinate or view represents before choosing a formula or transformation; the diagram and units are part of the reasoning.

1. Volume as layers of cross-sectional area

Every right prism can be viewed as identical cross-sectional layers. Rectangular prisms use a rectangular cross-section.

Text-equivalent representation: Prism text model: identical 6 cm by 4 cm rectangular layers stacked through a perpendicular length of 3 cm; cross-sectional area 24 cm² multiplied by 3 cm gives 72 cm³.

Decision check: Apply this lesson's rule to the worked example: Volume = area of constant cross-section × perpendicular prism length. Identify the exact value, label or property that satisfies the rule.

2. Triangular prisms and volume reasoning

For a triangular prism, calculate one-half base times perpendicular height before multiplying by length. If the cross-section stays fixed, volume changes by the same factor as prism length.

Text-equivalent representation: Triangular-prism text model: two congruent triangular ends, each base 8 cm and perpendicular height 5 cm, joined by a prism length of 10 cm.

Decision check: Apply this lesson's rule to the worked example: Keep square units on the cross-section and cubic units on the finished volume. Identify the exact value, label or property that satisfies the rule.

Worked examples with complete reasoning

Worked example 1

Question: Find the volume of a cube with side 5 cm.

  1. Identify the constant cross-sectional area.
  2. Multiply it by the prism length to obtain 125 cubic units.

Answer: 125

Check: All right prisms use cross-sectional area × length.

Worked example 2

Question: Find the volume of a rectangular prism 6 cm by 4 cm by 3 cm.

  1. Volume = length × width × height.
  2. 6 × 4 × 3 = 72 cm³.

Answer: 72

Check: Volume uses cubic units.

Worked example 3

Question: A right prism has volume 180 cm³ and cross-sectional area 30 cm². Find its length.

  1. Use volume = cross-sectional area × length.
  2. 180 ÷ 30 = 6 cm.

Answer: 6

Check: Divide volume by cross-sectional area.

Method map for this topic

  1. Volume as layers of cross-sectional area: Volume = area of constant cross-section × perpendicular prism length.
  2. Triangular prisms and volume reasoning: Keep square units on the cross-section and cubic units on the finished volume.

Common mistakes and how to correct them

  • rectangular prism volume: Not yet. Volume uses cubic units. Recheck the rectangular prism volume step, then try again.
  • triangular prism volume: Not yet. Find the cross-sectional area first, then multiply by prism length. Recheck the triangular prism volume step, then try again.
  • missing prism length: Not yet. Divide volume by cross-sectional area. Recheck the missing prism length step, then try again.
  • prism structure: Not yet. All right prisms use cross-sectional area × length. Recheck the prism structure step, then try again.
  • volume units: Not yet. A square-unit cross-section multiplied by a length gives cubic units. Recheck the volume units step, then try again.
  • volume scaling: Not yet. Identify which dimension changes and which cross-sectional measurements stay fixed. Recheck the volume scaling step, then try again.

Practice

  1. Find the volume of a cube with side 5 cm.
  2. Find the volume of a rectangular prism 2 m by 3 m by 4 m.
  3. Find the volume of a rectangular prism 6 cm by 4 cm by 3 cm.
  4. Find the volume of a rectangular prism 10 cm by 2.5 cm by 4 cm.
  5. A right prism has volume 180 cm³ and cross-sectional area 30 cm². Find its length.
  6. A right prism has volume 336 cm³ and cross-sectional area 48 cm². Find its length.

<details> <summary>Answers and reasoning</summary>

  1. 125 — Identify the constant cross-sectional area. Multiply it by the prism length to obtain 125 cubic units.
  2. 24 — Identify the constant cross-sectional area. Multiply it by the prism length to obtain 24 cubic units.
  3. 72 — Volume = length × width × height. 6 × 4 × 3 = 72 cm³.
  4. 100 — Volume = length × width × height. 10 × 2.5 × 4 = 100 cm³.
  5. 6 — Use volume = cross-sectional area × length. 180 ÷ 30 = 6 cm.
  6. 7 — Use volume = cross-sectional area × length. 336 ÷ 48 = 7 cm.

</details>

Checklist and vocabulary

| Skill | Observable evidence | | --- | --- | | identify prism cross section length | Identify a right prism's constant cross-section and perpendicular length. | | calculate rectangular triangular prism volume | Calculate rectangular and triangular prism volumes in cubic units. | | reason about right prism volumes | Solve and explain right-prism volume problems involving a missing length, cubic units or a scale change. |

Retrieval prompts

  • State the defining relationship for each method above without looking.
  • Give one example where two visually similar representations mean different things.
  • Diagnose one misconception from the list and explain the first corrective step.
  • Name a check that is independent of simply repeating the same calculation.

Teach-back and self-check

Close the worked examples and explain volumes of rectangular and triangular prisms as if you were helping another Year 7 student. Begin by naming the clue that tells you which relationship, representation or operation is needed. Then show one complete example with labels and units where they matter. At each line, explain what stayed equivalent and why the next step is allowed.

Use this four-part check before calling the method secure:

  1. Identify: What information is given, and what must be found or justified?
  2. Represent: Which number line, diagram, table, expression, graph or organised list makes the relationship visible?
  3. Reason: Which rule connects the representation to the required answer?
  4. Verify: Can you check by estimation, substitution, an inverse operation, a second representation or the original context?

If the explanation depends on “I just knew”, return to the method map and name the missing decision. A durable method is one you can explain, check and use in a changed context after a delay.

Transfer task

Sketch or describe a real object or layout that uses volumes of rectangular and triangular prisms. Label every necessary dimension, coordinate, axis or view, solve the stated problem, and name one limitation of the representation.

Sources