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.
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.
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=area of the cross-section×length
Volume and capacity (how much a container holds) are linked: 1 cm³ holds 1 millilitre, so 1000 cm³ holds 1 litre.
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.
Make your prediction first. Help opens once you've locked it in.
PredictRead the problem and decide what you think before you try anything. A wrong guess is fine: it shows you what to look for.
A triangular prism
What is the area of the triangle end?
TryTry it on the board and compare the result with your prediction.
Build the prism
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.
Try it yourselfYour edits are saved in this browser.
Build the prism up to 5 cm. What is each layer, and the whole volume? Was your prediction right?
Use the board. This ticks when you get there.
InvestigateWork out why it happens, one step at a time.
Cross-section times length
Here is the rule for any prism, and capacity.
Worked exampleBefore each step, say what you think comes next.
A=21×6×4=12 cm2Find the area of the cross-section (a triangle: halve).
V=12×10=120 cm3Times the length: one layer of 12 for each centimetre.
120 cm3=120 mLCapacity: 1 cm³ holds 1 mL.
1000 cm3=1 L1000 mL is a litre: a 10 cm cube holds 1 L.
Now try one: a triangle end with base 6 cm and height 4 cm, 10 cm long.
Read Lena's working. What shape is the tent's end?
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?
Lena finds how much air is inside a tent. Click the line where her working goes wrong.
ModifyChange the problem on purpose and see what happens.
One change at a time
Each task changes one thing.
Need a hint?
1 L = 1000 cm³.
Another hint
Try the same number for all three lengths.
Try it yourselfYour edits are saved in this browser.
Make a box that holds exactly 1 litre.
Use the board. This ticks when you get there.
A container holds 2500 cm³. How many litres is that?
A drink carton is 6 cm by 6 cm by 20 cm. How many millilitres does it hold?
A prism's cross-section is a parallelogram with base 5 cm and height 2 cm. The prism is 4 cm long. Volume, in cm³?
PractiseA few on your own, to check you have it.
Prisms
A prism's cross-section has area 15 cm² and the prism is 8 cm long. Volume, in cm³?
How many cm³ is 3 litres?
A tent's triangle end has base 3 m and height 2 m. It is 4 m long. Volume, in m³?
Review: write 250 cm in metres.
MakeMake your own problem using what you have learned. Expect a few tries; that is normal.
A 24 cm³ wedge
Then find one that is as long as possible.
Try it yourselfYour edits are saved in this browser.
Need a hint?
The triangular prism is half a box. What volume must the box be?
Another hint
l × w × h = 48.
Build a triangular prism with volume 24 cm³.
Use the board. This ticks when you get there.
For teachers
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.
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.
Running the lesson
Predict: students who chose 60 found the box, not the prism.
Run: ask what the dashed rectangle on the floor shows.
Investigate: hold up a triangular box or a tent picture and ask what a slice looks like.
Modify: for the edge case, turn a real prism over.
Make: ask why l × w × h must be 48.
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.
Stretch and support
Stretch: A swimming pool is a prism whose cross-section is a trapezium (shallow to deep end). Estimate how many litres it holds.
Support: Use real prisms and slices of play dough; write the cross-section area first, every time.
Where it leads
Next: Circles (AC9M7M03).
Where it leads: volume of cylinders (Year 9), where the cross-section is a circle, and surface area of prisms.
What each step is for
a triangular prism: Committing to a volume for a triangular prism shows whether they halve for the triangle or multiply the three lengths as for a box.
build the prism: Building the triangular prism up shows each layer is the triangle's area, half of the matching box's layer.
cross-section times length: V = cross-section area × length; Lena forgets the triangle's half. The cross-section is the face that repeats all the way along, not the biggest face.
one change at a time: 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.
prisms: A triangular prism, a capacity conversion and a box in mL, mixed; forgetting the half is the planted slip; the review keeps units fresh.
a 24 cm³ wedge: Building a triangular prism with a set volume means the box it is half of must be 48: the halving works backwards.