Cube and Cuboid: Concepts, Formulas, and Clear Examples

Cube and cuboid questions ask about faces, edges, vertices, painting, cutting into smaller blocks, and opposite-face logic when a net is folded. A cube has 6 faces, 8 vertices, 12 edges; opposite faces never touch in a valid net.

MCQs often involve n cuts parallel to faces producing $(n+1)^3$ small cubes (or cuboid grid), counting cubes with 0, 1, 2, or 3 painted faces after surface dip. Visualize one corner and one body diagonal—formulas then scale.

Basic counts

Cube edges

$$12$$

Cube vertices

$$8$$

Cube faces

$$6$$

• Cuboid with dimensions $a \times b \times c$: volume $abc$; surface area $2(ab+bc+ca)$.

Opposite faces on nets

In standard dice/cube nets, faces separated by one square in a ‘cross’ are often opposite. Mark pairs before folding mentally.

• Adjacent faces in the net share an edge in the solid.
• Maximum three faces meet at one vertex.

Painted cube after division

Unit cubes from n equal cuts per edge

$$N = (n+1)^3$$

3 faces painted (corners)

$$8 \text{ (always for full outer dip)}$$

2 faces painted (edge middles, not corners)

$$12(n-1)$$

1 face painted (face centers)

$$6(n-1)^2$$

0 faces painted (interior)

$$(n-1)^3$$

Example — 3×3×3 painted cube

$n=2$ cuts → 27 cubes. 3-face: 8; 2-face: $12(1)=12$; 1-face: $6(1)=6$; 0-face: 1. Sum = 27 ✓

Cuboid cut counts

If edges divided into $a+1$, $b+1$, $c+1$ pieces, interior unpainted count is $(a-1)(b-1)(c-1)$ when $a,b,c \ge 1$ dip variants—read whether all outer faces painted.

Diagonal and shortest path on surface

• Unfold faces; shortest path becomes straight line on net.
• Space diagonal of cube side $s$: $s\sqrt{3}$.

Exam traps

• Confusing cuts (n) with pieces (n+1) per edge.
• Forgetting inner cubes only when all six faces painted before cutting.

Step-by-step method for exam questions

1. Identify solid type (cube vs cuboid) and edge division count.
2. If painted, classify unit cubes by how many outer faces they had.
3. Apply corner/edge/face/interior formulas.
4. For nets, label opposite pairs before answering adjacency.
5. Sum categories to verify total piece count.
6. Select option matching verified total.

Formula cheat sheet

Frequently asked questions

Do formulas work if only one face painted?

No—standard counts assume entire outer surface painted before cutting.

How to find opposite face in MCQ net?

Skip one square rule in 4-cross net; or fold mentally two steps.

Cuboid vs cube painted counts?

Use generalized edge/face formulas; corners still 8 if all dimensions > 1.

Practice on this platform

Use 4 practice sets (10 MCQs, detailed solution after each answer) and 2 timed mocks (20 minutes) on this topic to convert formulas into speed and accuracy.