Maths
What is the net of a 3D shape, and how do you know if a net makes a cube?
Quick answer
A net is a flat 2D pattern that folds up into a 3D shape. A cube's net is 6 squares joined edge to edge, and a cube has 11 different nets.
In one sentence
A net is what you get when you unfold a 3D shape and lay it flat: a 2D pattern that folds back up into the solid.
Explained
Take an empty cereal box, carefully open the glued flaps and press it flat on the table. The flat piece of card you are holding is the box's net. Every face of the box is still there, joined to its neighbours along the fold lines. Fold it back up and you have the box again.
box
scissors
glue
The faces of a 3D shape are flat 2D shapes, so a net is made of 2D shapes too. To work out what a net looks like, count the faces of the solid and name their shapes (see faces, edges and vertices):
| 3D shape | Faces in its net |
|---|---|
| Cube | 6 squares |
| Cuboid (rectangular prism) | 6 rectangles, in 3 matching pairs |
| Triangular prism | 2 triangles and 3 rectangles |
| Square-based pyramid | 1 square and 4 triangles |
| Triangular pyramid (tetrahedron) | 4 triangles |
| Cylinder | 2 circles and 1 rectangle |
| Cone | 1 circle and 1 sector (a "pizza slice" of a circle) |
A shape can have more than one net, because you can cut it open along different edges. A cube has 11 different nets. Two nets count as the same if one is just the other turned round or flipped over. A cuboid whose length, width and height are all different has 54 nets.
Not every set of 6 squares makes a cube. When you fold it, two squares might land on the same face and leave a hole somewhere else. Three quick checks:
- No more than 4 squares in a straight line. A fifth square would wrap round onto the first.
- No 2 × 2 block of squares. Four squares in a block cannot fold up.
- Opposite faces never touch. In a straight row of three squares, the first and the third end up opposite each other.
A sphere is the odd one out: it has no exact net. You cannot flatten a ball without stretching or tearing it, which is why every flat map of the Earth is slightly distorted.
Nets are also the easiest way to find surface area: the area of the net is the surface area of the shape.
Examples
- The cross. A row of 4 squares with one square above and one below the second square. It is the best-known cube net.
- The staircase. Squares in steps of two: 2, then 2, then 2, each pair shifted one square along. It folds into a cube too.
- A cylinder's net. For a tin with radius 3 cm and height 8 cm, the rectangle is 8 cm tall and as long as the circle's circumference: 2 × π × 3 ≈ 18.8 cm.
- Not a net. Six squares in a 2 × 3 rectangle contain 2 × 2 blocks, so they cannot fold into a cube.
Common mistakes
- Counting the wrong number of faces. A triangular prism needs 5 faces, not 6. Fix: count the faces of the solid first, then draw.
- Thinking any 6 squares make a cube. Fix: check for a row of 5 or more, a 2 × 2 block, and two squares that would fold onto the same face.
- Making the cylinder's rectangle too short. Its length must equal the circumference of the circle (2πr), or the ends will not fit.
- Counting turned or flipped copies as new nets. Fix: cut them out. If one fits exactly on top of the other, it is the same net.
Step by step
Step by step
- Count the facesName the shape and list its faces, for example a cube: 6 squares.
- Draw the middle rowWith a ruler, draw a row of 4 equal squares, say 5 cm each. These become the four sides.
- Add the top and bottomDraw one square above and one below the same square of the row.
- Add tabsDraw small flaps on 7 of the outer edges so you have something to glue.
- Cut, fold and glueCut round the outside, fold every line the same way, then glue the tabs inside.
Why 7 tabs? A cube has 12 edges. In the net, 5 of them are already joined by folds, so 12 − 5 = 7 edges still need gluing.
Practice questions
Nets of 3D shapes
Try each one, then check your answer.
- How many faces are in the net of a cuboid?
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A cuboid has 6 faces, so its net has 6 rectangles. - Which shapes make the net of a triangular prism?
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2 triangles and 3 rectangles - Which 3D shape folds up from 4 equal triangles?
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A triangular pyramid (tetrahedron) - A net has 2 hexagons and 6 rectangles. What 3D shape does it make?
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A hexagonal prism - How many faces are in the net of a pentagonal pyramid?
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1 pentagon + 5 triangles = 6 faces - Six squares in one straight row: is it a net of a cube?
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No. The ends would wrap round and overlap. A cube net never has more than 4 squares in a row. - Six squares in a 2 × 3 rectangle: is it a net of a cube?
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No. It contains 2 × 2 blocks, which cannot fold up. - A row of 4 squares with one square above and one below the second square: is it a cube net?
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Yes, it is the cross net. - A row of 4 squares with two more squares both above the row: is it a cube net?
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No. Both extra squares fold onto the top, so the bottom is missing. - In the cross net, the row is numbered 1, 2, 3, 4 and squares 5 and 6 are above and below square 2. Which square is opposite square 1?
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In a row, squares two apart are opposite, so square 3. - A cube net is made of squares with sides of 6 cm. What is its area?
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6 × 6 × 6 = 216 cm² - What is the area of the net of a cuboid 5 cm × 3 cm × 2 cm?
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2 × (15 + 10 + 6) = 2 × 31 = 62 cm² - A cylinder has radius 7 cm. How long is the rectangle in its net? Use π ≈ 3.14.
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2 × 3.14 × 7 = 43.96 cm - Every cube net has 14 outer edges. What is the perimeter of a cube net with 4 cm squares?
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14 × 4 = 56 cm
Word problems
Nets in real life
Try each one, then check your answer.
- You make a cube-shaped gift box with 10 cm edges. How much card do you need, not counting tabs?
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6 × 10 × 10 = 600 cm² - A tissue box is 24 cm long, 12 cm wide and 10 cm tall. How much card is its net, without tabs?
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2 × (288 + 240 + 120) = 2 × 648 = 1,296 cm² - A tin has radius 4 cm and height 11 cm. What size is the paper label that wraps round it? Use π ≈ 3.14.
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Length 2 × 3.14 × 4 = 25.12 cm, so 25.12 cm by 11 cm (about 276 cm²). - On a dice, opposite faces add up to 7. You write 3 at one end of a row of three squares in a dice net. What goes at the other end?
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The ends of a row of three are opposite, so 7 − 3 = 4. - A tent is a triangular prism 2 m long. Each end is a triangle with base 1.6 m and height 0.6 m, and each sloping side is 1 m wide. How much fabric covers the two ends and two sloping sides?
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Ends: 2 × (1.6 × 0.6 ÷ 2) = 0.96 m²; sides: 2 × 1 × 2 = 4 m²; total 4.96 m²
Questions people ask
What is a net of a 3D shape?
It is the flat shape you get by unfolding a solid along some of its edges. Folded back up along the lines, it makes the 3D shape with no gaps and no overlaps.
How many nets does a cube have?
Exactly 11. Each one is made of 6 squares, and turned or flipped versions of the same pattern are not counted again.
How do you know if a net will make a cube?
It needs 6 squares, no row longer than 4, and no 2 × 2 block. Then imagine folding it: if two squares land on the same face, it is not a net. Cutting it out and folding it is the surest test.
What is not a net of a cube?
Any pattern that leaves a gap or overlaps when folded, such as 6 squares in a line, a 2 × 3 rectangle, or a row of 4 with both extra squares on the same side.
What is the net of a cylinder and a cone?
A cylinder unfolds into 2 circles and a rectangle. A cone unfolds into 1 circle and a sector of a bigger circle.
Does a sphere have a net?
No. A curved ball cannot be laid flat without stretching, so it has no exact net. Footballs and globes get close by using many small flat panels or curved strips.
How do you make a net of a cube with paper?
Draw the cross with a ruler (a row of 4 squares, plus one above and one below), add 7 glue tabs, cut it out, fold every line and glue the tabs.
Quick quiz
Nets of 3D shapes
Check what you know with 4 quick questions.
Question 1 of 4
How many squares are in the net of a cube?
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Related topics
3D shapes
3D shapes are solid shapes with length, width and height, like cubes, spheres, cylinders and pyramids. They have faces, edges and vertices.
Surface area
Surface area is the total area of all the outside faces of a 3D shape. For a cube with side a, it is 6a², because a cube has six equal square faces.
2D shapes
2D shapes are flat shapes with length and width but no thickness, like circles, triangles, squares and hexagons. You can draw them on paper.
Volume
Volume is the amount of space a 3D object takes up, measured in cubic units like cm³ or m³. A box 4 cm by 3 cm by 2 cm has a volume of 24 cm³.
Faces, edges and vertices
Faces are the flat surfaces of a 3D shape, edges are the lines where two faces meet, and vertices are the corners. A cube has 6 faces, 12 edges and 8 vertices.
Symmetry
A shape has symmetry when one half is a mirror image of the other (line symmetry), or when it looks the same after being turned (rotational symmetry).