Nets of Solids

Cut along some edges of a box and open it flat: the connected flat faces are its net. A net must show every face once, with shared edges matching so it folds back without overlap.

Parents: ask your child to name the face shapes before guessing the solid. In the lab, the fold result is checked from the shared-edge arrangement, not from an animation that merely looks plausible.

By the end you will recognise and draw the required P4 solid representations, identify foldable nets and name the solid made by a given net.

Recognise the solid representation

Use the faces and the way they meet.

• A cube is drawn with six equal square faces; the back edges may be dashed.
• A cuboid with unequal edge lengths is drawn with six rectangular faces, with matching opposite faces. Say this exact phrase when a cube is another option.
• A triangular prism has two congruent triangular ends joined by three rectangular sides. The end can be a 3-4-5 right triangle; the prism length is a separate edge.
• A square pyramid has one square base and four triangular faces meeting at one apex.
• A cylinder has two circular ends and one curved side.
• A cone has one circular base and one curved side narrowing to an apex.

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Solid and net lab: count faces, then fold

🤔 Predict first: A flat shape has six equal square faces arranged as a foldable net. What can it make?

A cubeA cylinderA cone

Text alternative: the lab's solid drawings name the face list beside each picture. The cube has six equal square faces and twelve edges; the cuboid has unequal labelled edges 4 cm, 3 cm and 2 cm; the prism has two triangular ends with sides 3 cm, 4 cm and 5 cm plus a separate 6 cm length; the pyramid has a square base and four triangles; the cylinder and cone descriptions use circles and curved sides.

Draw the required 2D representations

These are drawing procedures, so use a ruler and keep the far edges clear.

Cube: draw a front square with four equal labelled sides. Draw a second equal square shifted up and to one side. Join corresponding corners; use dashed lines for the hidden back edges. The six faces have twelve edges, all representing equal lengths.

Cuboid: draw a front rectangle, for example 4 cm by 2 cm. Draw a matching rectangle shifted up and to one side, then join corresponding corners. Label the third edge 3 cm. Opposite rectangular faces match; the unequal edge lengths mean it is a cuboid with unequal edge lengths, not a cube.

Triangular prism: draw a 3-4-5 triangle, draw a congruent triangle a fixed distance away, then join corresponding vertices. The two triangles are the ends and the three joins are rectangular sides. Keep the prism length label separate from 3, 4 and 5.

Square pyramid: draw a square base, mark one apex above it, join the apex to all four base corners. Use a dashed line for a far base edge or hidden apex edge when the view requires it. The triangular faces meet at the one apex.

Face checks for a drawing

$$6\text{ equal squares}\rightarrow\text{cube}$$

$$2\text{ matching triangles}+3\text{ rectangles}\rightarrow\text{triangular prism}$$

$$1\text{ square}+4\text{ triangles}\rightarrow\text{square pyramid}$$

Identify a cube net by folding

A cube net has six squares, but six squares alone are not enough. Fold each shared edge mentally: each square must land on a different side of the cube. The classic valid case is a row of four squares with one square above and one below the second square. A six-square strip is invalid because faces repeat a side when folded. A row with tabs at both ends can also put two faces on the same side.

Fold check

$$6\text{ squares is necessary, not sufficient}$$

$$\text{each fold must give a different cube side}$$

$$\text{no overlapping faces}\Rightarrow\text{valid net}$$

Nets of cuboids, prisms and pyramids

A cuboid net has four side rectangles in a strip and two matching rectangular bases. For an unequal cuboid, use side dimensions 4 cm, 3 cm and 2 cm: the strip's side rectangles use the 4-by-2 and 3-by-2 edges, and the two 4-by-3 bases attach to matching 4 cm edges.

A triangular-prism net has three rectangles in a strip and two congruent triangles attached to matching 4-unit edges. The triangles have sides 3, 4 and 5 and the strip length is 6. It folds to 2 triangles + 3 rectangles.

A square-pyramid net has one square base and four outward triangles, one attached to each square edge. The triangles meet at one apex when folded. The flat triangle height is a net length; do not replace it with an unrequired square-root or sloping-edge calculation.

Cone and cylinder recognition

These two representation checks are part of the P4 recognition breadth. A cylinder opens into two circles and a rectangle. A cone opens into one circle and one curved sector. No circumference, π, percentage or sector-angle calculation is needed.

Compare the curved solids

$$2\text{ circles}+1\text{ rectangle}\rightarrow\text{cylinder}$$

$$1\text{ circle}+1\text{ curved sector}\rightarrow\text{cone}$$

Practise with worked steps

Practice givens 1: A representation has two matching triangular ends, three rectangular side faces and a separate prism-length edge. Identify the solid and count its faces.

Reveal solution: Practice 1

Given two matching triangular ends, three rectangular side faces and a separate prism-length edge, the solid is a triangular prism. It has 2 triangular faces and 3 rectangular faces.

Count the prism faces

$$2\text{ triangular ends}+3\text{ rectangular sides}$$

$$\Rightarrow\text{triangular prism}$$

Practice givens 2: A net has a row of four unit squares, with one square joined above and one below the second square. Decide whether it is a valid cube net.

Reveal solution: Practice 2

Given a row of four unit squares with one square joined above and one below the second square, fold the four-row strip around and the two tabs over the remaining sides. It is a valid cube net because six different cube sides are covered once.

Fold the cube net

$$4+1+1=6\text{ squares}$$

$$\text{six different sides and no overlap}\Rightarrow\text{valid}$$

Practice givens 3: A net has one square base and four triangles attached outward, one on each square edge. Identify the solid it forms.

Reveal solution: Practice 3

Given one square base and four triangles attached outward, one on each edge, the triangles meet at one point when folded. The formed solid is a square pyramid.

Identify the pyramid

$$1\text{ square base}+4\text{ triangles}$$

$$\Rightarrow\text{square pyramid}$$

Watch out

Quick recap

🎯 Mastery check

Answer all 8 — your progress is saved on this device.

1. Which solid has six equal square faces?

   Cuboid with unequal edge lengthsCubeTriangular prism

2. A solid has labelled edges 4 cm, 3 cm and 2 cm and rectangular faces. What is it?

   A cubeA square pyramidA cuboid with unequal edge lengths

3. What faces make a triangular prism?

   1 square and 4 triangles2 triangles and 3 rectangles6 equal squares

4. What faces make a square pyramid?

   2 triangles and 3 rectangles2 circles and 1 rectangle1 square and 4 triangles

5. Which arrangement is required for a valid cube net?

   Any six connected squaresSix squares that fold to six different sidesSix squares that overlap when folded

6. A net has two circles and one rectangle. Which solid does it form?

   ConeCylinderSquare pyramid

7. A net has one circle and one curved sector. Which solid does it form?

   CylinderCuboidCone

8. A six-square strip is proposed as a cube net. What should you do?

   Accept it because six faces always workReject it because folding repeats sides and overlaps facesCall it a cone

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