Find the volume, the lateral surface area and the total surface area of the cuboid whose dimensions are:
(i) length = 12 cm, breadth = 8 cm and height = 4.5 cm
(ii) length = 26 m, breadth = 14 m and height = 6.5 m
(iii) length = 15m, breadth = 6 m and height = 5 dm
(iv) length = 24m, breadth = 25 cm and height = 6 m

Answer

To solve the problem of finding the volume, lateral surface area, and total surface area of a cuboid given its dimensions, we will follow these steps for each part of the question.

### Formulas to Use:
1. **Volume of a Cuboid (V)**:
\[ V = \text{length} \times \text{breadth} \times \text{height} \]

2. **Lateral Surface Area (LSA)**:
\[ LSA = 2h(l + b) \]
where \( h \) is the height, \( l \) is the length, and \( b \) is the breadth.

3. **Total Surface Area (TSA)**:
\[ TSA = 2(lb + bh + hl) \]

### Step-by-Step Solutions:

#### (i) Length = 12 cm, Breadth = 8 cm, Height = 4.5 cm

1. **Volume**:
\[ V = 12 \times 8 \times 4.5 = 432 \, \text{cm}^3 \]

2. **Lateral Surface Area**:
\[ LSA = 2 \times 4.5 \times (12 + 8) = 2 \times 4.5 \times 20 = 180 \, \text{cm}^2 \]

3. **Total Surface Area**:
\[ TSA = 2 \times (12 \times 8 + 8 \times 4.5 + 4.5 \times 12) = 2 \times (96 + 36 + 54) = 2 \times 186 = 372 \, \text{cm}^2 \]

#### (ii) Length = 26 m, Breadth = 14 m, Height = 6.5 m

1. **Volume**:
\[ V = 26 \times 14 \times 6.5 = 2366 \, \text{m}^3 \]

2. **Lateral Surface Area**:
\[ LSA = 2 \times 6.5 \times (26 + 14) = 2 \times 6.5 \times 40 = 520 \, \text{m}^2 \]

3. **Total Surface Area**:
\[ TSA = 2 \times (26 \times 14 + 14 \times 6.5 + 6.5 \times 26) = 2 \times (364 + 91 + 169) = 2 \times 624 = 1248 \, \text{m}^2 \]

#### (iii) Length = 15 m, Breadth = 6 m, Height = 5 dm (0.5 m)

1. **Convert Height**:
\[ 5 \, \text{dm} = 0.5 \, \text{m} \]

2. **Volume**:
\[ V = 15 \times 6 \times 0.5 = 45 \, \text{m}^3 \]

3. **Lateral Surface Area**:
\[ LSA = 2 \times 0.5 \times (15 + 6) = 2 \times 0.5 \times 21 = 21 \, \text{m}^2 \]

4. **Total Surface Area**:
\[ TSA = 2 \times (15 \times 6 + 6 \times 0.5 + 0.5 \times 15) = 2 \times (90 + 3 + 7.5) = 2 \times 100.5 = 201 \, \text{m}^2 \]

#### (iv) Length = 24 m, Breadth = 25 cm (0.25 m), Height = 6 m

1. **Convert Breadth**:
\[ 25 \, \text{cm} = 0.25 \, \text{m} \]

2. **Volume**:
\[ V = 24 \times 0.25 \times 6 = 36 \, \text{m}^3 \]

3. **Lateral Surface Area**:
\[ LSA = 2 \times 6 \times (24 + 0.25) = 2 \times 6 \times 24.25 = 291 \, \text{m}^2 \]

4. **Total Surface Area**:
\[ TSA = 2 \times (24 \times 0.25 + 0.25 \times 6 + 6 \times 24) = 2 \times (6 + 1.5 + 144) = 2 \times 151.5 = 303 \, \text{m}^2 \]

### Summary of Results:
1. For (i): Volume = 432 cm³, LSA = 180 cm², TSA = 372 cm²
2. For (ii): Volume = 2366 m³, LSA = 520 m², TSA = 1248 m²
3. For (iii): Volume = 45 m³, LSA = 21 m², TSA = 201 m²
4. For (iv): Volume = 36 m³, LSA = 291 m², TSA = 303 m²