Prism problems

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.

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.

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.

Predict: Read the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
Try: Try it on the board and compare the result with your prediction.
Investigate: Work out why it happens, one step at a time.
Modify: Change the problem on purpose and see what happens.
Practise: A few on your own, to check you have it.
Make: Make your own problem using what you have learned. Expect a few tries; that is normal.

PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.

How deep?

Change the litres to cubic metres first.

A tank has a base 4 m by 3 m and holds 36 000 L when full. How deep is it?0.3 m3 m12 m3000 m

TryTry it on the board and compare the result with your prediction.

Build the tank

Try it yourselfYour edits are saved in this browser.

Build the 4 m by 3 m tank that holds 36 000 L. Was your prediction right?

Use the board. This ticks when you get there.

InvestigateWork out why it happens, one step at a time.

The end, then the length

Here is the routine.

Worked exampleBefore each step, say what you think comes next.

$A = \tfrac{1}{2} \times 6 \times 4 = 12\ \text{cm}^2$ Find the area of the end with its own formula.
$V = A \times h = 12 \times 10 = 120\ \text{cm}^3$ Times the length of the prism.
$h = V \div A$ To find a missing length, divide the volume by the end's area.

Now try one.

Read Mia's working. What shape is the end?

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?

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

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 sidesThe 3 cm and 4 cm sides (they meet at the right angle)Only the 5 cm sideThe 5 cm side and the length

ModifyChange the problem on purpose and see what happens.

Work backwards

Need a hint?

The end's area times the length must be 60.

Another hint

Try legs 3 and 4 (end area 6).

Need a hint?

48 000 L = 48 m³.

Another hint

The base is 6 × 4 = 24 m².

Try it yourselfYour edits are saved in this browser.

Make a triangular prism with a volume of 60 cm³.

Use the board. This ticks when you get there.

A box has a volume of 240 cm³ and a base 8 cm by 6 cm. How tall is it, in cm?

A pool is 6 m by 4 m and must hold 48 000 L. Set its depth.

Use the board. This ticks when you get there.

A concrete step is 2 m long. Its end is an L made of two rectangles: 0.6 m by 0.3 m and 0.3 m by 0.5 m. How many m³ of concrete does it need?

PractiseA few on your own, to check you have it.

Prism problems

A prism's end has an area of 15 cm² and it is 8 cm long. What is its volume in cm³?

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

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

Review: how many mL does 1.25 L hold?

MakeMake your own problem using what you have learned. Expect a few tries; that is normal.

A box to order

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

Try it yourselfYour edits are saved in this browser.

Need a hint?

960 mL = 960 cm³.

Another hint

With a base of 80 cm², the height is 960 ÷ 80.

Design a box with a base area of 80 cm² that holds 960 mL.

Use the board. This ticks when you get there.

For teachers

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.

Open the teacher guide: answers, solutions, slides and a worksheet.

Students will be able to

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.

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.

Running the lesson

Predict: ask what to do with the litres first.
Run: ask how the depth would change for a bigger base.
Investigate: ask Mia which face stays the same all the way along.
Modify: ask for the depth before they move the slider.
Make: ask for the plan in one sentence.

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.

Stretch and support

Stretch: A box must hold 1 L. Which whole-number dimensions use the least card? Compare at least four designs.

Support: Give a printed prism with the end shaded, and a formula card V = A × h.

Where it leads

This completes AC9M8M02.

Where it leads: cylinders (Year 9), surface area of prisms, and rates of filling (Rates).

What each step is for

how deep?: Committing to a depth from a capacity shows whether they convert litres to cubic metres and divide by the base area.
build the tank: Building a tank to a capacity shows the depth that a base of 12 m² needs.
the end, then the length: Any prism is end area times length (Mia uses l × w × h on a triangle); only the end's own measurements count.
work backwards: Missing lengths from volumes and capacities, and a prism whose end is a composite shape.
prism problems: Volume, capacity and a missing length, mixed; using l × w × h on a triangle is the planted slip; the conversion review keeps litres fresh.
a box to order: Designing to both a base area and a capacity makes the student plan the length from V = Ah.

Now you can

Find the volume of any right prism from the area of its end.
Find a missing length from a volume or a capacity.
Answer in the units the problem asks for.

Practise this skillNew questions, new numbers each timeGet 3 right in a row to mark it practised.Next lesson16. Area of a circleCut a circle into thin slices and lay them top to tail: they make a shape close to a rectangle, πr long and r high. So the area of a circle is A = πr². It is between 2r² and 4r², about 3.14 times r².

Source: content/maths/measurement/prism-problems/lesson.md