EQN-04-12

Nets & 3-D Visualisation

Match solids to their unfolded nets and count faces, edges, and vertices — spatial-reasoning staples.

~5 min read•Difficulty 2/5

A net is an unfolded solid

A net is the 2-D pattern that folds up into a 3-D solid. The number of regions in the net equals the number of faces.

Cube net : 6 squares Cuboid net : 6 rectangles (3 matching pairs)

Cylinder net : 2 circles + 1 rectangle

Faces, edges, vertices

Cube : 6 faces, 12 edges, 8 vertices

Cuboid : 6 faces, 12 edges, 8 vertices

Square pyramid : 5 faces, 8 edges, 5 vertices

Euler's relation holds for these solids: faces $+$ vertices $-$ edges $= 2.$

Folding logic

On a cube net, squares on opposite sides of the cross-shaped pattern end up opposite each other; adjacent squares share an edge. Use this to match a net to its painted cube.

Surface area from the net

Add the areas of all the regions in the net — that IS the surface area, with no formula to memorize.

Diagram

A cube unfolds into a net of 6 equal squares — handy for surface area.

Worked examples

Example 1 (difficulty 1/5)

How many edges does a cube have?

◎ Predict the answer first:

Example 2 (difficulty 2/5)

A cube has edge 4. What is its surface area, found from its net?

◎ Predict the answer first:

Tips

Number of net regions $=$ number of faces of the solid.
Surface area $=$ sum of the net’s region areas — no separate formula needed.
On a cube net, squares opposite in the pattern fold to opposite faces.

Common mistakes

Confusing edges, faces, and vertices counts

Why: Three similar-sounding counts.

Fix: Cube: 6 faces, 12 edges, 8 vertices — memorize the trio.

Choosing a net with the wrong number of faces

Why: Any cross-like shape looks plausible.

Fix: A cube net must have exactly 6 squares arranged so none overlap when folded.

← Back to Geometry Practice this lesson →