Nets of Three-Dimensional Figures

Have you ever taken apart a cardboard box and laid it completely flat on the floor? If so, you have created a net.

A net is a two-dimensional (flat) pattern that can be folded along its edges to create a three-dimensional (solid) figure. Recognizing and drawing nets helps us understand the surface area and structure of 3D shapes.

Common 3D Figures and Their Nets

Different 3D figures unfold into different types of 2D nets based on their faces.

Cube

A cube is made of 6 identical square faces.

• A valid net for a cube will always have exactly 6 equal squares.
• There are 11 different ways to arrange 6 squares so that they fold perfectly into a cube. The most recognizable is the "cross" shape (four squares in a row, with one square on each side of the middle).

Rectangular Prism

A rectangular prism has 6 rectangular faces. The opposite faces are identical.

• For a rectangular prism with dimensions $3 \times 2 \times 1$, its net will consist of 3 pairs of matching rectangles.
• You will have two $3 \times 2$ rectangles, two $3 \times 1$ rectangles, and two $2 \times 1$ rectangles.

Triangular Prism

A triangular prism has 5 faces: 2 identical triangular bases and 3 rectangular sides.

• Its net usually looks like three rectangles joined side-by-side, with a triangle attached to the top and bottom of the middle rectangle.

How to Tell if a Net Works

To determine if a flat pattern will fold into a closed 3D shape, you can use mental math to visualize folding it:

1. Pick a base: Choose one face on the net to be the bottom of your shape.
2. Fold up the sides: Imagine folding the adjacent faces up by $90^\circ$.
3. Check for overlaps and gaps: If two faces fold onto the exact same side, they overlap, and your net is invalid. If there is a missing side (a gap), the net is also invalid.

Example Problems

Example 1: Which net folds into a cube? If you see a net with 5 squares in a straight row, it cannot be a cube. When you fold it, the ends will overlap, and the sides will be left open. A valid cube net must have a central row of at most 4 squares, with the remaining squares acting as the top and bottom flaps.

Example 2: Identify the 3D shape from a net. If a net shows a large square surrounded by four triangles attached to its edges, what does it form? Answer: When you fold the four triangles upward so their points meet at the top, it forms a square pyramid.