Mr. Ward's Teaching Materials

021i - Composite Prism Volumes

10 min read

Lesson 21 — Composite and Right Prism Volumes

Strand: Measurement | Descriptor: AC9M7M02 | Duration: 45 minutes

Learning Intentions

  • To calculate the volume of a prism whose cross-section is a composite shape.
  • To decompose a composite solid into simpler prisms.

Success Criteria

I can:

  1. Identify the constant cross-section of a composite prism.
  2. Find the cross-sectional area by splitting it into rectangles and triangles.
  3. Calculate volume as cross-sectional area × length, or by adding/subtracting simpler prisms.
  4. Choose the more efficient of the two strategies and justify the choice.

Warmup

(6 minutes — retrieval relay, mini whiteboards)

  1. Triangular prism: cm, cm, cm. Find .
  2. Rectangular prism: cm. Find .
  3. A prism has cross-sectional area and length cm. Find .
  4. A prism has volume and length cm. Find its cross-sectional area.

(Answers: ; ; ; .)

Bridging question: In Q3, did you need to know the shape of the cross-section? What does that tell you about how to handle a more complicated one?

Activities

Activity 1 — Explicit Instruction: the Cross-section Method (14 min)

The central principle, stated once and used throughout:

This holds for any prism, however complicated the cross-section. So a composite prism is really a composite area problem (Lesson 17) followed by one multiplication.

The protocol:

  1. Find the face that stays constant along the solid — that is the cross-section.
  2. Split that face into rectangles and triangles.
  3. Calculate the total cross-sectional area.
  4. Multiply by the length.

I do — the “house” prism (a shed). The cross-section is a rectangle m wide and m tall, with a triangular roof of base m and height m on top. The shed is m long.

Identifying the cross-section — the key difficulty. Rotate a physical model (or a slide) so the shed lies on its side. The cross-section is not whichever face happens to be at the front; it is the face that repeats identically all the way through. Ask: “If I sliced this like a loaf of bread, which direction gives identical slices?”

We do — the “L-shaped” prism. The cross-section is an L made from a rectangle and a rectangle; the prism is cm long.

You do: Five composite prisms from a worksheet, at least two of which require a missing cross-section length to be deduced first.

Activity 2 — Add or Subtract Whole Prisms (8 min)

An alternative route: treat the solid as prisms joined or removed.

I do — a block with a rectangular channel cut through it. A solid block cm by cm by cm has a rectangular channel cm long, cm wide and cm deep cut along its full length.

Cross-check by the cross-section method: the cross-section is an rectangle minus a rectangle, giving , and ✓

Having students verify one solid both ways is the single best self-check available in this topic.

Choosing a method — rule of thumb:

If the same cross-section runs the whole length, use . If separate blocks of different lengths are stuck together, add whole prisms.

You do: A step-shaped solid made from a block with a block on top. (Answer: .)

Activity 3 — Inquiry: the Swimming Pool (10 min)

Pairs.

A swimming pool is m long and m wide. Its floor slopes evenly: the depth is m at the shallow end and m at the deep end.

  1. Sketch the pool’s cross-section along its length. What shape is it?
  2. Find the cross-sectional area.
  3. Find the volume of water when the pool is full.
  4. Given litres, how many litres does it hold?

Socratic scaffolding:

PromptPurpose
Understand: which direction gives identical slices?Slicing across the width gives identical m-wide slices, so the cross-section is the side view.
What shape is that side view?A trapezium — parallel sides m and m, m apart.
Have you seen a related problem?Yes — split it, exactly as in Lesson 17.
How would you split it?A rectangle , plus a triangle of base and height .
Carry it out (area).
Now the volume. What is the “length” here?The pool’s width, m — the direction the cross-section travels.
Carry it out (volume) litres.
Looking back — sanity checkAn “average depth” of m over gives ✓

Discussion: The averaging shortcut works here only because the floor slopes evenly. Ask students when it would fail. (If the floor had a step or a curve, the average of the two end depths would not equal the true average depth.)

Checks for Understanding

(5 minutes — exit ticket)

  1. A prism has a cross-section made from a rectangle and a triangle (, ), and is cm long. Find its volume.
  2. A cm block has a cm channel cut along its length. Find the remaining volume.
  3. A shed’s cross-section has area and the shed is m long. Find its volume.
  4. Reasoning. Explain how to identify the cross-section of a prism when it is not the face you happen to be looking at.
  5. A prism has volume and a cross-sectional area of . Find its length.

Answers: 1. , so ; 2. ; 3. ; 4. Look for the direction in which every slice is identical — the “loaf of bread” test. The cross-section is perpendicular to the length; 5. cm.

Common Misconceptions

MisconceptionHow to pre-empt it
Assuming the cross-section is always the front-facing rectangle.Use physical models and rotate them. Apply the loaf-of-bread test aloud every time.
Using the wrong measurement as the “length”.The length is the direction perpendicular to the cross-section. In the pool problem it is the width, m — a deliberate trap.
Adding cross-sectional areas of parts that have different lengths.The method needs one constant cross-section. If lengths differ, add whole prisms instead.
Double-counting an overlap when splitting the cross-section.Draw dashed split lines on the diagram and label each region exactly once.
Forgetting to halve for a triangular part of a composite cross-section.Require the formula, not just the numbers, on every working line.
Using for volume.Area is the intermediate step, volume the final one — the exponents differ. Check the units at each stage.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A cube of side cm has a square hole of side cm drilled straight through from one face to the opposite face. What is the remaining volume?

Answer

E2 (AMC Junior style). A prism’s cross-section is a trapezium with parallel sides cm and cm, and perpendicular distance cm between them. The prism is cm long. Find its volume.

Answer

Split the trapezium into a rectangle and a triangle , :

(Equivalently, .)

E3 (Challenge). Two identical triangular prisms, each with cross-sectional area and length cm, are glued together along their rectangular faces to form a single prism. What is the total volume? Does gluing change the answer?

Answer

Each has volume , so the total is . Gluing changes the shape and the surface area, but not the volume — matter is neither added nor removed.

E4 (Challenge). A concrete kerb is a prism m long. Its cross-section is a rectangle cm wide and cm tall, with a triangular chamfer of base cm and height cm cut from one top corner. Find the volume of concrete in cubic metres.

Answer

Work in centimetres first:

Since , the volume is .

Homework

  1. A prism has a cross-section made from a rectangle and a triangle (, ), and is cm long. Find its volume.
  2. A prism has cross-sectional area and length cm. Find its volume.
  3. A cm block has a cm channel cut along its full length. Find the remaining volume, and verify your answer using the cross-section method.
  4. A garden shed has a cross-section made from a rectangle m by m with a triangular roof of base m and height m. The shed is m long. Find its volume.
  5. A step-shaped solid is made from a cm block with a cm block on top. Find the total volume.
  6. A water trough has a trapezium cross-section: parallel sides cm and cm, with cm between them. It is m long. Find its capacity in litres.
  7. Reasoning. Explain why the volume of a prism does not depend on how you choose to split its cross-section.
  8. Challenge. A composite prism has volume and length cm. Its cross-section is a rectangle of width cm with a triangle of base cm sitting on top. If the rectangle is cm tall, find the height of the triangle.

Answers: Q1 — , . Q2 — . Q3 — ; cross-section , ✓. Q4 — , . Q5 — . Q6 — ; litres. Q8 — ; rectangle , so triangle ; gives cm.

Graph View

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