Nets of 3D Shapes

What is a net?

A net is a flat (2D) pattern that folds up along its edges to make a 3D shape. Imagine carefully cutting open a cardboard cereal box along some of its edges and flattening it out on the table — the flat shape you are left with is a net of that box. Fold it back up and you get the box again.

Nets are useful because they let us see every face of a 3D shape laid out flat, all at once. This is exactly how cardboard boxes, gift boxes and packaging are designed and printed: as flat nets that machines fold and glue into solid shapes.

Before we look at nets, it helps to know the three parts of a 3D shape:

• A face is a flat surface (each face appears in the net).
• An edge is the straight line where two faces meet.
• A vertex is a corner where edges meet (the plural is vertices).

For a closer look at how 2D and 3D shapes are related, see our lesson on 2D and 3D shapes.

The net of a cube

A cube (like a dice or a sugar cube) has 6 faces, and every face is a square of the same size. It also has 12 edges and 8 vertices.

So the net of a cube is made of 6 equal squares joined together. The classic example is a "cross" or "plus sign" shape: a row of four squares, with one extra square above the second one and one below it. When you fold the four squares in the row into a loop, the top square folds over to become the lid and the bottom square folds under to become the base — and a cube appears!

Interestingly, there are 11 different nets that all fold into a cube. They each use 6 squares, just arranged in different ways.

Nets of other common shapes

Different 3D shapes have different nets, because they have different faces.

• Cuboid (a box shape, like a brick or a cereal box) — has 6 rectangular faces, so its net is 6 rectangles. Opposite faces are equal pairs.
• Triangular prism (like a Toblerone bar) — has 2 triangular ends and 3 rectangular faces joining them. Its net is 2 triangles plus 3 rectangles.
• Square-based pyramid (like an Egyptian pyramid) — has 1 square base and 4 triangular faces that meet at a point. Its net is 1 square in the middle with a triangle on each side.
• Cylinder (like a tin of beans) — has 2 circular ends and 1 curved surface. The curved surface unrolls into a rectangle, so its net is 2 circles and 1 rectangle.
• Cone (like an ice-cream cone) — has 1 circular base and a curved surface that unrolls into a part of a circle (a "fan" or sector). Its net is 1 circle and 1 sector.

A good rule of thumb: a prism always has two identical ends joined by rectangles, and a pyramid always has one base with triangles climbing to a single point.

A diagram in words

Picture the net of a square-based pyramid lying flat on the table.

In the centre there is a single square — this will become the base. On each of its four edges, a triangle is attached, pointing outward, so the whole thing looks like a four-pointed star or a square with a triangular flap sticking out of every side.

Now imagine lifting the four triangles up off the table at the same time, swinging them up on their hinges (the edges of the square). The tips of all four triangles rise and meet at a single point high above the centre of the square. Snap — you have built a pyramid, with the square as its base and the four triangles as its sloping sides.

Worked example 1

A 3D shape has a net made of 6 rectangles, where the rectangles come in three matching pairs. What is the shape?

Six rectangular faces, with opposite faces equal, describes a cuboid (a box shape). If all six rectangles were equal squares instead, it would be a cube.

Worked example 2

You are told a net is made of 2 circles and 1 rectangle. What 3D shape does it fold into, and what do the parts become?

This is the net of a cylinder. The two circles become the flat top and bottom (the circular ends), and the rectangle rolls around to form the curved side. The length of the rectangle is exactly the distance around the circle (its circumference).

Worked example 3

How many faces, edges and vertices does a triangular prism have? Use its net to count.

The net has 2 triangles and 3 rectangles, which is 5 faces in total. Counting carefully, a triangular prism has 5 faces, 9 edges and 6 vertices. The net makes the faces easy to count because they are all laid out flat in front of you.

Why this matters

Nets connect the flat, 2D world of drawings to the solid, 3D world of real objects — a hugely important skill. Designers, architects and engineers constantly switch between flat plans and 3D models, and packaging companies design every box you have ever opened as a net first. Nets also make it easy to work out the surface area of a solid: because all the faces are laid out flat, you can simply find the area of each face and add them up. So learning nets now is the first step toward calculating how much wrapping paper covers a present or how much metal makes a tin can. It trains your "spatial reasoning" — the ability to picture how things fit together in space — which is one of the most useful skills in maths, science and everyday life.

Activity: become a box designer

You need squared paper, scissors, a ruler and some sticky tape.

1. Draw the cross net of a cube: four squares in a row, with one square above and one below the second square. Make each square 4 cm by 4 cm.
2. Cut it out, fold along every line, and tape the edges. Did it form a perfect cube? Check that every face is a square with no gaps.
3. Experiment: can you find a different arrangement of 6 squares that also folds into a cube? Try a "T" shape or a staircase shape. See how many of the 11 possible cube nets you can discover.
4. Trap test: draw 6 squares all in a single straight line and try to fold it. What goes wrong? (You will find some faces overlap.) This shows that the arrangement matters, not just the number of squares.
5. Challenge: unfold a small real box (like a teabag box) along its edges and draw its net. Then design and build your own gift box for a small object, starting from a net you draw yourself.