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# Prism problems

Area, volume, circles and Pythagoras · Volume

<!-- Speaker note: Students use V = Ah for rectangular and triangular prisms, work backwards to missing lengths, and solve capacity problems in context (AC9M8M02, node 2). About 40 minutes (Predict and Run 8 min · Investigate 10 min · Modify 10 min · Practise and Make 12 min). -->

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## Today

**We are learning to** the volume of any right prism is the area of its end (the cross-section) times its length: V = Ah. Work backwards to find a missing length: divide the volume by the area of the end. Give the answer in the units the problem needs.

- I can find the volume of any right prism from the area of its end.
- I can find a missing length from a volume or a capacity.
- I can answer in the units the problem asks for.

<!-- Speaker note: Before class: Do the Run board yourself first. Students need volume and capacity conversions (previous lesson) and composite areas. The one idea: V = A × h for every right prism, where A is the area of the end. The common mistake is using length × width × height on a prism that is not a box (Mia's mistake). The board's prism is built in layers of the end's area; its triangular prism has a right-triangle end. Wider: packaging (the box with the least card for a given volume), and finding dimensions for a required capacity. -->

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## Why it matters

> A farmer needs a water trough that holds 500 L, a builder orders concrete in cubic metres, a pool needs filling. Each is a prism problem, often run backwards.

<!-- Speaker note: A **right prism** has the same cross-section (its end, or base) all the way along. For every right prism, $$V = A \times h$$ where $A$ is the area of the end and $h$ is the length (or height) of the prism. Find $A$ with the right area formula for the end: a rectangle, a triangle, or a composite shape. To find a missing length, work backwards: $h = V \div A$. Convert units before you start, so every length is in the same unit. -->

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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 tank has a base 4 m by 3 m and holds 36 000 L when full. How deep is it?**

- 0.3 m
- 3 m
- 12 m
- 3000 m

<!-- Speaker note: What happens: 3 m. 36 000 L = 36 m³. The base is 4 × 3 = 12 m², so the depth is 36 ÷ 12 = 3 m. Take two or three predictions before anyone runs it. -->

<!-- Speaker note: Teacher: ask what to do with the litres first. -->

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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%2Fprism-problems%2Fstep-2&code=widget%3A+prism%0Abase%3A+rectangle%0Aslider+l%3A+1%2C+6%2C+2%0Aslider+w%3A+1%2C+6%2C+2%0Aslider+h%3A+1%2C+6%2C+1%0Aunits%3A+m%0Acapacity%3A+on%0A&fresh=1}}

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

<!-- Speaker note:  -->

<!-- Speaker note: Teacher: ask how the depth would change for a bigger base. -->

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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?

* 3 m. 36 000 L = 36 m³. The base is 4 × 3 = 12 m², so the depth is 36 ÷ 12 = 3 m.

<!-- 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 routine.

Now try one.

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

1) $V = A \times h = 12 \times 10 = 120\ \text{cm}^3$ — Times the length of the prism.
2) $h = V \div A$ — To find a missing length, divide the volume by the end's area.

<!-- 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 = 12cm^2 and say "Find the area of the end with its own formula." Line 1: write V = A × h = 12 × 10 = 120cm^3 and say "Times the length of the prism." Line 2: write h = V ÷ A and say "To find a missing length, divide the volume by the end's area." -->

<!-- Speaker note: Teacher: ask Mia which face stays the same all the way along. -->

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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 triangular prism's end has legs 5 cm and 4 cm (a right angle between them). The prism is 9 cm long. What is its volume in cm³?**

* Answer: 90

<!-- Speaker note: A = ½ × 5 × 4 = 10 cm², so V = 10 × 9 = 90 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

**Mia finds the volume of a triangular prism. Click the line where her working goes wrong.**

```
 1  A prism's end is a right triangle with legs 3 cm and 4 cm. The prism is 10 cm long.
 2  Mia uses volume = length × width × height.
 3  V = 3 × 4 × 10 = 120 cm³.
```

* Answer: line 2

<!-- Speaker note: Line 2: l × w × h is for a rectangular box. Any prism is the area of its end times its length: ½ × 3 × 4 × 10 = 60 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>

## What you need

**The end of a triangular prism is a right triangle with sides 3 cm, 4 cm and 5 cm. Which do you need for its area?**

- All three sides
- The 3 cm and 4 cm sides (they meet at the right angle)
- Only the 5 cm side
- The 5 cm side and the length

* Answer: The 3 cm and 4 cm sides (they meet at the right angle)

<!-- Speaker note: The two legs are a base and its perpendicular height: ½ × 3 × 4 = 6 cm². -->

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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>

## Work backwards



- **A volume of 60:** 
- **Find the height:** 
- **Fill a pool:** 
- **A composite end:** 

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

<!-- Speaker note: What this step is for: Missing lengths from volumes and capacities, and a prism whose end is a composite shape. Hints, in order: The end's area times the length must be 60. / Try legs 3 and 4 (end area 6). / 48 000 L = 48 m³. / The base is 6 × 4 = 24 m². For students who need more support: Give a printed prism with the end shaded, and a formula card V = A × h. For early finishers: A box must hold 1 L. Which whole-number dimensions use the least card? Compare at least four designs. The hard part, on purpose: Deciding which face is the end. Ask students to point to the shape that stays the same all the way along before they calculate. When students get stuck: A student multiplies three lengths for a triangular prism. Ask: What shape is the end, and what is its area? Fix: V = area of the end × length. / A student mixes m and cm in one problem. Ask: Are all your lengths in the same unit? Fix: Convert first, then calculate. / A student multiplies when working backwards. Ask: What times the end's area gives the volume? Fix: Divide the volume by the end's area. -->

<!-- Speaker note: Teacher: ask for the depth before they move the slider. -->

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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>

## End area times length

**A prism's end has an area of 15 cm² and it is 8 cm long. What is its 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>

## A fish tank

**A fish tank is 50 cm by 40 cm by 30 cm. How many litres does it hold?**

* Answer: 60

<!-- Speaker note: 50 × 40 × 30 = 60 000 cm³ = 60 L. -->

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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>

## Buckets

**A trough is 2 m long, 1 m wide and 0.5 m deep. How many 10 L buckets fill it?**

* Answer: 100

<!-- Speaker note: 2 × 1 × 0.5 = 1 m³ = 1000 L, and 1000 ÷ 10 = 100 buckets. -->

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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: litres

**Review: how many mL does 1.25 L hold?**

* Answer: 1250

<!-- Speaker note: 1.25 × 1000 = 1250 mL. -->

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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 box to order

Then explain how you found the height before you moved the slider.

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

<!-- Speaker note: What this step is for: Designing to both a base area and a capacity makes the student plan the length from V = Ah. Hints, in order: 960 mL = 960 cm³. / With a base of 80 cm², the height is 960 ÷ 80. -->

<!-- Speaker note: Teacher: ask for the plan in one sentence. -->

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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 tank with a 2 m by 1.5 m base holds 4500 L. How deep is the water?**

* 4500 L = 4.5 m³; 4.5 ÷ 3 = 1.5 m.

<!-- 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: This completes AC9M8M02. Where it leads: cylinders (Year 9), surface area of prisms, and rates of filling (Rates). -->