1. Cuboid Dimension Reassembly

What is this?

Small cubes are joined together to make a bigger cube or cuboid.

Questions usually ask:

• New surface area
• New volume
• Surface area lost after joining

---

Easy Rules

✅ Volume never changes.

Total Volume Before = Total Volume After

✅ Surface area decreases because touching faces become hidden.

---

Example

Question

Eight unit cubes (1 × 1 × 1) are joined to make one 2 × 2 × 2 cube.

Find

• New surface area
• Surface area lost

---

Step 1: Surface area before joining

Each small cube has

6 faces

Area of one cube

6 × 1² = 6

There are

8 cubes

So

Total Surface Area

= 8 × 6

= 48 square units

---

Step 2: Make one big cube

The new cube is

2 × 2 × 2

ASCII

      ______
     /_____/|
    /_____/ |
    |     | |
    |     | /
    |_____|/

Surface Area

= 6 × side²

= 6 × 2²

= 6 × 4

= 24 square units

---

Step 3: Find lost area

Some faces are now inside the cube.

They cannot be seen.

Lost Area

= Old Surface Area

− New Surface Area

= 48 − 24

= 24 square units

---

Final Answer

New Surface Area = 24

Lost Surface Area = 24

---

Easy Picture

Before joining

□  □

□  □

□  □

□  □

Every cube has all faces visible.

---

After joining

      ______
     /_____/|
    /_____/ |
    |     | |
    |     | /
    |_____|/

Many faces touch each other.

Those touching faces disappear from the outside.

---

Quick Formula

Surface Area

Cube

= 6a²

Volume

Cube

= a³

Lost Area

Old Surface Area

−

New Surface Area

---

2. Axis-Based Slicing

What is this?

A cuboid is cut into unit cubes.

Questions ask:

• Number of cuts
• Number of cubes formed

---

Example

A cuboid measures

Length  = 5

Breadth = 3

Height  = 2

---

Step 1: Draw the cuboid

        __________
       /_________/|
      /_________/ |
      |         | |
      |         | /
      |_________|/

---

Step 2: Cuts along Length

Length = 5

|----|----|----|----|

5 blocks

Need 4 cuts

Formula

Length − 1

= 5 − 1

= 4

---

Step 3: Cuts along Breadth

Breadth = 3

|----|----|

3 blocks

Need 2 cuts

Formula

3 − 1

= 2

---

Step 4: Cuts along Height

Height = 2

|----|

2 blocks

Need 1 cut

Formula

2 − 1

= 1

---

Step 5: Total Cuts

Length Cuts

+

Breadth Cuts

+

Height Cuts

=

4 + 2 + 1

=

7 cuts

---

Step 6: Number of Cubes

Multiply all dimensions.

5 × 3 × 2

= 30 cubes

---

Final Answer

Cuts Needed

= 7

Unit Cubes

= 30

---

Easy Diagram

Original Cuboid

5 × 3 × 2

After slicing

□□□□□□□□□
□□□□□□□□□
□□□□□□□□□

30 small cubes

---

Quick Formulas

Number of Cubes

Length × Breadth × Height

---

Number of Cuts

(Length − 1)

+

(Breadth − 1)

+

(Height − 1)

---

Shortcut Table

Question	Formula
Number of unit cubes	L × B × H
Cuts along Length	L − 1
Cuts along Breadth	B − 1
Cuts along Height	H − 1
Total Cuts	(L−1) + (B−1) + (H−1)
Surface Area of Cube	6a²
Volume of Cube	a³
Lost Surface Area	Old SA − New SA

Memory Trick

JOINING CUBES
─────────────
✔ Volume stays same
✔ Surface area decreases

CUTTING CUBES
─────────────
✔ Cubes = L × B × H
✔ Cuts = (L−1)+(B−1)+(H−1)

These formulas are enough to solve almost all Cuboid & Cube Reassembly questions in aptitude exams.