Representing 3-d in 2-d
180. Unfolding Solids · From 3D shapes to flat patterns
A valid net of a cube has 6 congruent square faces arranged to fold into a closed solid.
A net of a 3D solid is a 2D pattern that folds along its edges to form the solid. A valid cube net has 6 connected square faces arranged so that folding produces a closed cube. There are 11 distinct nets for a cube.
What this lesson covers
Try to break it
Try folding fully, then back to flat. All four patterns — cross, T-shape, L-shape, and staircase — each fold into a closed cube with no gaps or overlaps. That is the definition of a valid net.
How you build it
Construct a cube net from six squares using the square tool.
- Draw a square in the centre — this is the base face.
- Attach a square to the top edge of the base.
- Attach a square to the bottom edge of the base.
- Attach a square to the left edge of the base.
- Attach a square to the right edge of the base.
- Attach the sixth square to the outer edge of the top square — the six squares form a cube net.
The proof, step by step
Prove that the net folds into a cube.
- Count the faces: A cube has 6 faces, so the net must have exactly 6 squares.
- Check congruence: All squares must be of equal size to fold into a regular cube.
- Verify folding: When folded, opposite faces must not overlap, and all edges must meet.
Worked example
Which of the following arrangements of 6 congruent squares can be folded to form a closed cube without any overlapping faces?
A cross shape (or similar T/L variations) with 6 squares allows each square to become a face of the cube. Rectangles, straight lines, or incomplete shapes cannot fold into a closed cube without gaps or overlaps.
- A 2x3 rectangle of squares
- A cross shape with 6 squares — correct
- A straight line of 6 squares
- An L-shape with 5 squares