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# Prisms and capacity

Area, volume, circles and Pythagoras · Volume

<!-- Speaker note: Students build triangular prisms in layers, find volume as cross-section area × length for any right prism, and link volume to capacity (1 cm³ = 1 mL, 1000 cm³ = 1 L) (AC9M7M02, node 2). About 40 minutes (Predict and Run 8 min · Investigate 12 min · Modify 10 min · Practise and Make 10 min). -->

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## Today

**We are learning to** any right prism has the same shape all the way along: its cross-section. Volume = area of the cross-section × length, for triangles too (halve first). Capacity: 1 cm³ holds 1 mL, and 1000 cm³ is a litre.

- I can find the cross-section of a prism.
- I can find the volume of a triangular prism.
- I can change between cm³, mL and litres.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need V = A × h for boxes (previous lesson) and triangle area (Area unit). The one idea: every right prism is its cross-section repeated along its length, so V = cross-section area × length. The common mistake is forgetting the triangle's half (Lena's mistake, a spiral of the triangles lesson). Capacity conversions (mL, L) are new. -->

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## Why it matters

> A tent, a chocolate bar in a triangular box, a drink carton: how much do they hold? If the shape is the same all the way through, one rule finds them all.

<!-- Speaker note: A **right prism** has the same shape all the way through, like a loaf of bread: cut it anywhere, straight across, and you get the same slice. That slice is the **cross-section**. A box has a rectangle cross-section; a **triangular prism** (a tent, a triangular chocolate box) has a triangle. The layer idea works for every prism. Each layer is the cross-section, and the layers go all the way along: $$V = \text{area of the cross-section} \times \text{length}$$ Volume and **capacity** (how much a container holds) are linked: 1 cm³ holds 1 millilitre, so 1000 cm³ holds 1 litre. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Predict

**A prism's triangle end is half of a 4 cm by 3 cm rectangle. The prism is 5 cm long. What is its volume?**

- 60 cm³
- 12 cm³
- 30 cm³
- 6 cm³

<!-- Speaker note: What happens: 30 cm³. The triangle is ½ × 4 × 3 = 6 cm², and 5 layers of 6 is 30. 60 is the whole box; 6 is one layer. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: students who chose 60 found the box, not the prism. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Try it

{{embed: https://studio.learningwaypoints.com/tutorials/_playground-board/index.html?id=deck%2Fmaths%2Fmeasurement%2Fprisms%2Fstep-2&code=widget%3A+prism%0Abase%3A+triangle%0Al%3A+4%0Aw%3A+3%0Aslider+h%3A+0%2C+6%2C+1%0Aunits%3A+cm%0A&fresh=1}}

[Open this step in the tutorial](https://studio.learningwaypoints.com/tutorials/maths/measurement/prisms/?full=step-2#step-2)

<!-- Speaker note: The board stands the prism on its triangle end, so its length points up (h on the board). The orange triangle is the cross-section: half of a 4 by 3 rectangle, cut along its diagonal. -->

<!-- Speaker note: Teacher: ask what the dashed rectangle on the floor shows. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## What happened?

* 30 cm³. The triangle is ½ × 4 × 3 = 6 cm², and 5 layers of 6 is 30. 60 is the whole box; 6 is one layer.

<!-- Speaker note: Reveal after running. Ask who predicted it, and what made them think so. -->

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<span style="background:#dbeafe; color:#1a1a1a; border:1.5px solid #3b82f6; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">💬 Discussion</span>

## Any prism

Here is the rule for any prism, and capacity.

Now try one: a triangle end with base 6 cm and height 4 cm, 10 cm long.

$$A = \tfrac{1}{2} \times 6 \times 4 = 12 \text{ cm}^2$$

1) $V = 12 \times 10 = 120 \text{ cm}^3$ — Times the length: one layer of 12 for each centimetre.
2) $120 \text{ cm}^3 = 120 \text{ mL}$ — Capacity: 1 cm³ holds 1 mL.
3) $1000 \text{ cm}^3 = 1 \text{ L}$ — 1000 mL is a litre: a 10 cm cube holds 1 L.

<!-- Speaker note: Model this live on the board before revealing anything: write each line and say why, pausing to ask the class what comes next. Then click to reveal the lines and compare. Script. Start: write A = 1 ÷ 2 × 6 × 4 = 12 cm^2 and say "Find the area of the cross-section (a triangle: halve)." Line 1: write V = 12 × 10 = 120 cm^3 and say "Times the length: one layer of 12 for each centimetre." Line 2: write 120 cm^3 = 120 mL and say "Capacity: 1 cm³ holds 1 mL." Line 3: write 1000 cm^3 = 1 L and say "1000 mL is a litre: a 10 cm cube holds 1 L." -->

<!-- Speaker note: Teacher: hold up a triangular box or a tent picture and ask what a slice looks like. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Any prism

**A prism's cross-section is a triangle with base 6 cm and height 4 cm. The prism is 10 cm long. What is its volume, in cm³?**

* Answer: 120

<!-- Speaker note: ½ × 6 × 4 = 12 cm², × 10 = 120 cm³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Find the mistake

**Lena finds how much air is inside a tent. Click the line where her working goes wrong.**

```
 1  A tent is a triangular prism 3 m long. Its triangle end has base 2 m and height 1.5 m.
 2  Area of the triangle end = 2 × 1.5 = 3 m²
 3  Volume = 3 × 3
 4  = 9 m³
```

* Answer: line 2

<!-- Speaker note: Line 2: the end is a triangle, so halve: ½ × 2 × 1.5 = 1.5 m². Volume = 1.5 × 3 = 4.5 m³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## The cross-section

**How do you find a prism's cross-section?**

- It is the biggest face
- It is always the face it stands on
- It is the face that is the same shape all the way along, like a slice of bread

* Answer: It is the face that is the same shape all the way along, like a slice of bread

<!-- Speaker note: Cut straight across anywhere: the cross-section is what every slice looks like. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## One change at a time



- **One litre:** Each task changes one thing.
- **Litres:** 
- **Millilitres:** 
- **A parallelogram:** 
- **Turned over:** 

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/prisms/?full=step-4#step-4)

<!-- Speaker note: What this step is for: One change at a time: a 1 litre box on the board, cm³ to litres, a carton in mL, a parallelogram cross-section, then a prism standing differently. Hints, in order: 1 L = 1000 cm³. / Try the same number for all three lengths. For students who need more support: Use real prisms and slices of play dough; write the cross-section area first, every time. For early finishers: A swimming pool is a prism whose cross-section is a trapezium (shallow to deep end). Estimate how many litres it holds. The hard part, on purpose: Spotting the cross-section of a prism lying on its side. Ask students to imagine slicing it like bread and describe the slice, rather than pointing to the face for them. When students get stuck: A student multiplies the three lengths of a triangular prism (Lena's mistake). Ask: What shape is the slice? What is its area? Fix: Halve for the triangle, then multiply by the length. / A student calls the face it stands on the cross-section. Ask: If you slice it like bread, what shape is each slice? Fix: The cross-section is the slice that repeats. / A student divides by 100 for litres. Ask: How many mL in a litre? Fix: 1000 mL = 1 L, and 1 cm³ = 1 mL. -->

<!-- Speaker note: Teacher: for the edge case, turn a real prism over. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Cross-section given

**A prism's cross-section has area 15 cm² and the prism is 8 cm long. Volume, in cm³?**

* Answer: 120

<!-- Speaker note: 15 × 8 = 120 cm³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Litres to cm³

**How many cm³ is 3 litres?**

* Answer: 3000

<!-- Speaker note: 3 × 1000 = 3000 cm³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## A tent

**A tent's triangle end has base 3 m and height 2 m. It is 4 m long. Volume, in m³?**

* Answer: 12

<!-- Speaker note: 3 m² × 4 = 12 m³. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Review: units

**Review: write 250 cm in metres.**

* Answer: 2.5

<!-- Speaker note: 250 ÷ 100 = 2.5 m. -->

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<span style="background:#fef3c7; color:#1a1a1a; border:1.5px solid #f59e0b; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✍️ Activity</span>

## Make: a 24 cm³ wedge

Then find one that is as long as possible.

[Open this step](https://studio.learningwaypoints.com/tutorials/maths/measurement/prisms/?full=step-6#step-6)

<!-- Speaker note: What this step is for: Building a triangular prism with a set volume means the box it is half of must be 48: the halving works backwards. Hints, in order: The triangular prism is half a box. What volume must the box be? / l × w × h = 48. -->

<!-- Speaker note: Teacher: ask why l × w × h must be 48. -->

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<span style="background:#dcfce7; color:#1a1a1a; border:1.5px solid #22c55e; padding:6px 18px; border-radius:8px; font-size:1em; font-weight:500;">✅ Check</span>

## Exit ticket

**A prism's triangle end has base 5 cm and height 4 cm; it is 6 cm long. Find its volume and its capacity in mL.**

* ½ × 5 × 4 = 10 cm², × 6 = 60 cm³ = 60 mL.

<!-- Speaker note: Students answer on paper or a mini whiteboard before the reveal. Anyone who cannot answer needs the main idea again next lesson. Where this leads: Next: Circles (AC9M7M03). Where it leads: volume of cylinders (Year 9), where the cross-section is a circle, and surface area of prisms. -->