Nets of 3D Shapes

Explore what nets are, how to identify the net of a 3D shape, and how to calculate surface area using nets.

What Is a Net?

A net is a flat (2D) shape that can be folded up to make a 3D shape. Imagine cutting along the edges of a cardboard box and laying it flat — that flat shape is the net.

Every face of the 3D shape appears exactly once in the net, and the faces must be connected along edges so they fold together correctly.

Note

Packaging designers use nets to create boxes and cartons. When you open a cereal box flat, you can see its net — six rectangles joined together.

Nets of Common Shapes

Cube: A cube has 6 identical square faces. There are 11 different valid nets for a cube. The most common looks like a cross of 4 squares with one square on each side.

Cuboid: A cuboid (rectangular box) has 6 rectangular faces in 3 opposite pairs. Its net contains 3 pairs of identical rectangles.

Triangular prism: Has 2 triangular faces and 3 rectangular faces — a total of 5 faces.

Square pyramid: Has 1 square base and 4 triangular faces — a total of 5 faces.

Identifying Valid Nets

Not every arrangement of faces makes a valid net. Rules for a valid net:

All faces must be present (correct number and shape)
When folded, no faces must overlap
Every face must be connected to at least one other face along a full edge

Tip

When checking a net, imagine folding it step by step. Start with the base, then fold each adjacent face up. Do the remaining faces land in the right places?

Surface Area From Nets

The surface area of a 3D shape is the total area of all its faces. Since a net shows all faces flat, you can calculate surface area by finding the area of each part of the net and adding them up.

Example: Find the surface area of a cube with side length $4 \text{ cm}$ .

Each face is a $4 \times 4$ square: area = $16 \text{ cm}^2$

A cube has 6 faces: $6 \times 16 = \mathbf{96 \text{ cm}^2}$

Example: A cuboid is $5 \text{ cm} \times 3 \text{ cm} \times 2 \text{ cm}$ . Find its surface area.

Top and bottom: $2 \times (5 \times 3) = 30 \text{ cm}^2$
Front and back: $2 \times (5 \times 2) = 20 \text{ cm}^2$
Left and right: $2 \times (3 \times 2) = 12 \text{ cm}^2$
Total: $30 + 20 + 12 = \mathbf{62 \text{ cm}^2}$

Warning

Surface area is measured in square units (cm², m²). Never write it without the squared unit symbol.

Real world: cereal box packaging

A breakfast cereal box is a cuboid 20 cm × 7 cm × 28 cm. Its net has six rectangles. Surface area $= 2(20 \times 7) + 2(20 \times 28) + 2(7 \times 28) = 280 + 1120 + 392 = 1792\text{ cm}^2$ . Packaging designers calculate this to know how much card stock to purchase for each box.

Real world: wrapping a gift

Wrapping a cube gift box of side 15 cm requires a sheet of paper at least as large as the net. Surface area $= 6 \times 15^2 = 1350\text{ cm}^2$ , so a sheet roughly 50 cm × 30 cm is needed (plus overlap for folding). Nets make it easy to estimate material needs.

Tip Draw out all the faces of the net and label each with its dimensions before calculating — it prevents counting a face twice or missing one entirely.

Practice Problems

How many faces does a triangular prism have?

Show answer

5 faces: 2 triangular faces and 3 rectangular faces.

Find the surface area of a cube with side length $3 \text{ cm}$.

Show answer

Each face: $3 \times 3 = 9 \text{ cm}^2$. Six faces: $6 \times 9 = 54 \text{ cm}^2$.