Nets of 3D shapes
🎯 In this topic you will
- Match 2D nets to the 3D shapes they form.
🧠 Key Words
- net
- tetrahedron
Show Definitions
- net: A 2D pattern of faces that can be folded along its edges to make a 3D shape.
- tetrahedron: A 3D shape with four triangular faces, four vertices, and six edges.
🧩 What a net means
A net of a 3D shape is what it looks like if it is opened out flat.
📦 A box made from card
Imagine you have a box made out of card.
✂️ Opening it out flat
If you cut along the edges you can open it up to make a flat shape.
🗺️ The flat shape has a name
This flat shape is called a net.
🧱 Folding it back into 3D
The net can be folded up again to make a box.
📦 Where nets are used
Nets are used in cardboard packaging.
🧾 How a box is made from a net
The net is drawn onto card, cut out and folded up to make a box.
📚 What you will learn next
In this section you will learn about the nets of 3D shapes.
❓ EXERCISES
$1$. Attif has cut and unfolded a box of cereal so that it makes a flat shape.
a. What shape was the box?
b. How many faces did the shape have?
c. What shape are the faces?
👀 Show answer
a. The box was a cuboid (rectangular prism).
b. It had $6$ faces.
c. The faces are rectangles.
$2$. Sara has cut and unfolded a packet of sweets so that it makes a flat shape.
a. What shape was the packet?
b. How many faces did the shape have?
c. What shape are the faces?
👀 Show answer
a. The packet was a hexagonal prism.
b. It had $8$ faces.
c. The faces are $2$ hexagons and $6$ rectangles.
❓ EXERCISES
🧠 Reasoning Tip
If more than one of the nets matches the correct number and shape of faces, try to visualise the faces folding up into the shape. The net should leave no gaps, and no faces should overlap when it is folded.
$3$. Marcus has an octagon-based pyramid.
Which of these is a net of Marcus’s shape?
👀 Show answer
Answer:C
An octagon-based pyramid has a net made from $1$ octagon (the base) and $8$ triangles (one for each side of the octagon).
Option C shows an octagon with triangles attached all the way around it, so it can fold up into an octagon-based pyramid without gaps or overlaps.
$4$. Identify the net of a pentagonal prism.
👀 Show answer
Answer:C
A pentagonal prism has $2$ pentagonal faces and $5$ rectangular faces.
Option C shows a strip of rectangles (the side faces) with two pentagons attached (the two ends), so it is the net of a pentagonal prism.
$5$. Name the $3$D shapes made with these nets.
👀 Show answer
a. Cone
b. Tetrahedron
c. Cylinder
d. Square-based pyramid
🧠 Think like a Mathematician
A triangle-based pyramid (tetrahedron) has 4 faces.
A square-based pyramid has 5 faces.
A pentagon-based pyramid has 6 faces.
Question: What link is there between the base shape (number of sides) and the number of faces for pyramids and prisms?
Equipment: Pen/pencil, paper (and a calculator if you want to check larger numbers).
Method:
- Read the three examples and record the base shape (number of sides) and the number of faces.
- Look for a pattern and continue the list for larger base polygons.
- Check your rule using a nonagon-based pyramid and a decagon-based pyramid.
- Repeat the same idea for prisms, then write a general rule.
Tasks:
A hexagon-based pyramid has faces.
A heptagon-based pyramid has .
An -based pyramid has .
Check this is true for a nonagon-based pyramid and a decagon-based pyramid.
👀 show answer
a.
- A hexagon-based pyramid has 7 faces.
- A heptagon-based pyramid has 8 faces.
- An octagon-based pyramid has 9 faces.
b. If the base has $n$ sides, the pyramid has $n$ triangular faces plus the base, so total faces = $n+1$.
- Nonagon-based pyramid: $n=9$ ⇒ faces = $9+1=10$.
- Decagon-based pyramid: $n=10$ ⇒ faces = $10+1=11$.
c. For an $n$-gonal prism, there are 2 bases and n rectangular side faces, so total faces = $n+2$.
d. General rule:
- Pyramids: base has $n$ sides ⇒ faces = $n+1$.
- Prisms: base has $n$ sides ⇒ faces = $n+2$.
❓ EXERCISES
$6$. You will be explaining to your partner how to work out what $3$D shape a net will make.
Take $1$ minute to prepare what you will say.
Explain to your partner.
Ask your partner to explain how they work out what $3$D shape a net will make.
👀 Show answer
Example explanation you could give:
“To work out what a net will make, I first look at the faces in the net and name their shapes. Then I count how many of each face there are, and I look for the face that could be the base. Next, I imagine folding along the edges: faces that touch in the net will touch on the solid. I check whether the faces can meet without overlapping, and whether the remaining face(s) would close the solid.”
- Identify face types: squares, rectangles, triangles, etc.
- Match to a known solid: e.g., one square + four triangles suggests a square-based pyramid.
- Check “closure”: do the side faces meet neatly around the base?
- Check connections: which edges will join when folded?
- Sanity check: does the net have the right number of faces for that solid?
Questions to ask your partner:
- “Which face do you choose as the base, and why?”
- “How do you check the faces will meet without overlapping?”
- “What clues tell you it is a prism or a pyramid?”