Find the volume, curved surface area and the total surface area of a cylinder having base radius 10.5 cm and height 18 cm.

Answer

To find the volume, curved surface area, and total surface area of a cylinder with a base radius of 10.5 cm and height of 18 cm, we will use the following formulas:

1. **Volume of a Cylinder (V)**:
\[ V = \pi r^2 h \]
2. **Curved Surface Area (CSA)**:
\[ CSA = 2 \pi r h \]
3. **Total Surface Area (TSA)**:
\[ TSA = 2 \pi r (r + h) \]

Where:
- \( r \) is the radius of the base of the cylinder,
- \( h \) is the height of the cylinder,
- \( \pi \) is approximately \( \frac{22}{7} \).

### Step 1: Calculate the Volume
1. Substitute the values into the volume formula:
\[ V = \pi r^2 h = \frac{22}{7} \times (10.5)^2 \times 18 \]

2. Calculate \( (10.5)^2 \):
\[ (10.5)^2 = 110.25 \]

3. Substitute \( 110.25 \) back into the volume formula:
\[ V = \frac{22}{7} \times 110.25 \times 18 \]

4. Calculate \( 22 \times 18 \):
\[ 22 \times 18 = 396 \]

5. Now substitute this value:
\[ V = \frac{396 \times 110.25}{7} \]

6. Calculate \( 396 \times 110.25 \):
\[ 396 \times 110.25 = 43671 \]

7. Finally, divide by 7:
\[ V = \frac{43671}{7} = 6237 \text{ cm}^3 \]

### Step 2: Calculate the Curved Surface Area
1. Substitute the values into the curved surface area formula:
\[ CSA = 2 \pi r h = 2 \times \frac{22}{7} \times 10.5 \times 18 \]

2. Calculate \( 2 \times \frac{22}{7} \):
\[ 2 \times \frac{22}{7} = \frac{44}{7} \]

3. Substitute this value:
\[ CSA = \frac{44}{7} \times 10.5 \times 18 \]

4. Calculate \( 10.5 \times 18 \):
\[ 10.5 \times 18 = 189 \]

5. Substitute back:
\[ CSA = \frac{44 \times 189}{7} \]

6. Calculate \( 44 \times 189 \):
\[ 44 \times 189 = 8316 \]

7. Finally, divide by 7:
\[ CSA = \frac{8316}{7} = 1188 \text{ cm}^2 \]

### Step 3: Calculate the Total Surface Area
1. Substitute the values into the total surface area formula:
\[ TSA = 2 \pi r (r + h) = 2 \times \frac{22}{7} \times 10.5 \times (10.5 + 18) \]

2. Calculate \( 10.5 + 18 \):
\[ 10.5 + 18 = 28.5 \]

3. Substitute back:
\[ TSA = 2 \times \frac{22}{7} \times 10.5 \times 28.5 \]

4. Calculate \( 2 \times \frac{22}{7} \):
\[ 2 \times \frac{22}{7} = \frac{44}{7} \]

5. Now substitute:
\[ TSA = \frac{44}{7} \times 10.5 \times 28.5 \]

6. Calculate \( 10.5 \times 28.5 \):
\[ 10.5 \times 28.5 = 298.25 \]

7. Substitute back:
\[ TSA = \frac{44 \times 298.25}{7} \]

8. Calculate \( 44 \times 298.25 \):
\[ 44 \times 298.25 = 13143 \]

9. Finally, divide by 7:
\[ TSA = \frac{13143}{7} = 1884.71 \text{ cm}^2 \approx 1881 \text{ cm}^2 \]

### Final Results:
- Volume = \( 6237 \text{ cm}^3 \)
- Curved Surface Area = \( 1188 \text{ cm}^2 \)
- Total Surface Area = \( 1881 \text{ cm}^2 \)