028i - Volume of Composite Prisms

Lesson 28 — Guided Practice: Volume of Composite Right Prisms

Strand: Measurement | Descriptor: AC9M8M02 | Duration: 45 minutes

Learning Intentions

• To decompose a composite right prism into two or more simple right prisms.
• To calculate the volume of composite solids by adding or subtracting component volumes.

Success Criteria

I can:

1. Identify how a composite solid’s cross-section can be split into simpler shapes.
2. Calculate the volume of each component using .
3. Add or subtract component volumes to find a total or remaining volume.
4. Solve a composite volume problem involving more than one cross-sectional shape.

Warmup

(5 minutes — bridge from 2D to 3D, mini whiteboards)

Recall the addition and subtraction strategies for composite 2D shapes.

1. An L-shaped floor plan can be split into two rectangles. If they have areas and , what is the total area?
2. A rectangle has a rectangular notch removed from one corner. Find the remaining area.
3. Predict: if you extruded each of these 2D shapes into a prism of length m, how would you find the volume of the whole composite solid?

Answers: 1. ; 2. ; 3. Prediction — most students will suggest finding the total (or remaining) cross-sectional area first, exactly as in Q1 and Q2, then multiplying by the length. Confirm this formally in Activity 1.

Activities

Activity 1 — Explicit Instruction: Composite Prisms by Addition (12 min)

The strategy. For a composite right prism, first find the total cross-sectional area by splitting it into simple shapes and adding their areas, then multiply the total by the prism’s length.

I do — a “house-shaped” prism. A cross-section is a rectangle wide and tall, topped with a triangle of base and height . The prism is long.

We do — an L-shaped cross-section, by addition. Split into two rectangles: and . Prism length cm.

You do:

1. A cross-section is a rectangle with a -base, -high triangle on top. Prism length cm.
2. An L-shaped cross-section splits into rectangles and . Prism length cm.

(Answers: 1. , ; 2. , .)

Activity 2 — Guided Practice: Composite Prisms by Subtraction (10 min)

The strategy. Some composite prisms are easier to see as a large simple prism with a smaller prism-shaped piece removed.

I do. A rectangular concrete beam in cross-section, long, has a rectangular channel (length width depth) cut all the way through it lengthwise for cabling.

We do. An L-shaped cross-section, this time by subtraction: a rectangle with a rectangular notch removed from one corner. Prism length cm.

You do:

1. A rectangular prism in cross-section, length , has a rectangular notch removed from a corner of its cross-section. Find the remaining volume.
2. Compare your answer for the L-shape in this activity’s “We do” with the L-shape in Activity 1’s “We do” (same overall outline, described differently). Are the two methods — addition and subtraction — describing the same physical shape?

Answers: 1. , . 2. Discussion point — both are valid descriptions of an L-shape, and either addition or subtraction can be used depending on which dimensions are given; students should confirm both methods give a sensible cross-sectional area for a genuine L-shape.

Activity 3 — Problem-solving: the Retaining Wall (13 min)

Pairs, then whole-class share.

A concrete garden retaining wall has a cross-section shaped like a trapezium: it is cm wide at the base, cm wide at the top, and cm tall. Along the base of the wall, a rectangular drainage channel wide and cm tall is cast into the wall (i.e. removed from the trapezium’s cross-section, along the full base). The wall is m long.

1. Sketch and label the cross-section.
2. Find the area of the trapezium.
3. Find the area of the drainage channel, and hence the remaining cross-sectional area.
4. Find the volume of concrete needed to build the wall, in .

Socratic scaffolding:

Prompt	Purpose
Understand: what is being removed, and from where?	A rectangular drainage channel is removed from along the base of the trapezium’s cross-section.
What units are the measurements given in, and what will you need before calculating volume?	All cross-section measurements are in cm, but the wall’s length is in m — units must be made consistent before the final multiplication.
Devise a plan	Find the trapezium’s area, subtract the channel’s area, then multiply by the length (converted to cm, or convert the final area to ).
Carry out — trapezium area	.
Carry out — channel and remaining area	; remaining .
Carry out — volume, with unit conversion	Length ; . Convert: .
Looking back — is a volume just under for a m long wall plausible?	Yes — the cross-section is quite thin (well under a tenth of a square metre), so a modest total volume over a m length is sensible.

Answers: 2. . 3. Channel ; remaining . 4. .

Checks for Understanding

(6 minutes — exit ticket, collected)

1. A cross-section is a rectangle with a -base, -high triangle on top. Find the total cross-sectional area.
2. The prism in Question 1 is cm long. Find its volume.
3. A rectangular cross-section has a notch removed from a corner. Find the remaining area.
4. The prism in Question 3 is cm long. Find its volume.
5. Reasoning. Explain how you would decide whether to use addition or subtraction to find a composite prism’s cross-sectional area.

Answers: 1. ; 2. ; 3. ; 4. ; 5. Use subtraction when the shape is most easily seen as a large simple shape with a piece missing (e.g. a notch or channel); use addition when the shape is most easily seen as two or more simple shapes placed together with no overlap — both methods are valid whenever the necessary dimensions are known, and sometimes either can be used for the same shape.

Common Misconceptions

Misconception	How to pre-empt it
Multiplying each component’s area by the length separately, then adding the volumes, but forgetting a component.	Model finding the total cross-sectional area first, then multiplying once by the length — one multiplication, not several.
Choosing addition when subtraction (or vice versa) is the more natural fit for the given dimensions.	Ask “is it easier to describe this as pieces put together, or as one big piece with something missing?” before starting.
Forgetting to convert all measurements to the same unit before calculating volume, especially when a length is given in metres and cross-section dimensions in centimetres.	Insist on a labelled “convert first” line, as modelled in Activity 3.
Subtracting the wrong shape’s area (e.g. using the outer trapezium’s parallel sides for the channel).	Require each component’s area to be calculated and labelled separately before any subtraction.
Reporting cross-sectional area with cubic units, or volume with square units.	Model the unit change explicitly: for area, (or ) for volume.

Enrichment — Competition-Style Problems

E1 (Kangaroo style). A composite prism’s cross-section is an L-shape formed from a rectangle with a rectangle removed from one corner. If the prism’s volume is , find its length.

Answer

E2 (AMC Junior style). A composite cross-section is a rectangle with a semicircular notch of radius cm removed from the middle of one long side. Find the remaining area, to decimal place, using .

Answer

E3 (Challenge). A composite prism’s cross-section is built by placing a -base, -height triangle on top of a rectangle, and the whole cross-section is doubled in every linear dimension. Find the new cross-sectional area, and compare it with the original.

Answer

Original: . Doubling every linear dimension multiplies every area component by : new area — confirmed by recomputing with doubled dimensions: .

E4 (Investigation — capstone). A swimming pool’s cross-section is a composite trapezoidal shape: it is m deep at the shallow end for a horizontal distance of m, then slopes down to m deep over the next m, then stays at m deep for a final m. The pool is m wide. Find the pool’s volume, in , by splitting the cross-section into a rectangle, a trapezium, and a rectangle.

Answer

Cross-section (a vertical slice along the pool’s length, depth vs. distance): rectangle ; trapezium ; rectangle . Total cross-section . Volume (using the pool’s width as the prism’s length).

Homework

1. A cross-section is a rectangle with a -base, -height triangle on top. Find the total area, and the volume for a prism length of cm.
2. A rectangular cross-section has a notch removed from one corner. Find the remaining area, and the volume for a prism length of cm.
3. An L-shaped cross-section splits into rectangles and . Find the total area, and the volume for a prism length of cm.
4. A concrete lintel has a cross-section that is a rectangle with a rebated channel removed along its length for a metal support. The lintel is m long. Find its volume in .
5. Reasoning. Explain why, for any composite prism, “find the total cross-sectional area first, then multiply by the length once” is more efficient and less error-prone than “find each component’s volume separately, then add or subtract the volumes.”
6. Challenge. A composite cross-section is a rectangle with a right-angled triangular notch (legs cm and cm) removed from one corner, and a further square notch removed from the opposite corner. Find the remaining area, and the volume for a prism length of cm.

Answers: Q1 — , . Q2 — , . Q3 — , . Q4 — cross-section ; length cm; . Q5 — the “total area first” method requires only one final multiplication, reducing the chance of arithmetic slips and avoiding the risk of forgetting to multiply one of several separate volumes by the length; both methods are mathematically equivalent by the distributive property, but the single-multiplication method is faster and easier to check. Q6 — ; .