Find the volume, curved surface area and the total surface area of a cylinder with diameter of base 14 cm and height 80 cm.

Answer

To find the volume, curved surface area, and total surface area of a cylinder with a diameter of 14 cm and a height of 80 cm, we will follow these steps:

### Step 1: Calculate the radius of the cylinder
The diameter of the base is given as 14 cm. The radius (r) is half of the diameter.

\[ r = \frac{\text{Diameter}}{2} = \frac{14 \text{ cm}}{2} = 7 \text{ cm} \]

### Step 2: Calculate the volume of the cylinder
The formula for the volume (V) of a cylinder is:

\[ V = \pi r^2 h \]

Substituting the values of \(\pi\), \(r\), and \(h\):

\[ V = \frac{22}{7} \times (7 \text{ cm})^2 \times 80 \text{ cm} \]

Calculating \(r^2\):

\[ (7 \text{ cm})^2 = 49 \text{ cm}^2 \]

Now substituting back into the volume formula:

\[ V = \frac{22}{7} \times 49 \text{ cm}^2 \times 80 \text{ cm} \]

Now, simplify:

\[ V = \frac{22 \times 49 \times 80}{7} \]

Since \(49\) can be divided by \(7\):

\[ 49 \div 7 = 7 \]

So we have:

\[ V = 22 \times 7 \times 80 \]

Calculating \(22 \times 7 = 154\):

\[ V = 154 \times 80 = 12320 \text{ cm}^3 \]

### Step 3: Calculate the curved surface area of the cylinder
The formula for the curved surface area (CSA) of a cylinder is:

\[ \text{CSA} = 2 \pi r h \]

Substituting the values:

\[ \text{CSA} = 2 \times \frac{22}{7} \times 7 \text{ cm} \times 80 \text{ cm} \]

The \(7\) in the numerator and denominator cancels out:

\[ \text{CSA} = 2 \times 22 \times 80 \]

Calculating:

\[ \text{CSA} = 44 \times 80 = 3520 \text{ cm}^2 \]

### Step 4: Calculate the total surface area of the cylinder
The formula for the total surface area (TSA) of a cylinder is:

\[ \text{TSA} = 2 \pi r (r + h) \]

Substituting the values:

\[ \text{TSA} = 2 \times \frac{22}{7} \times 7 \text{ cm} \times (7 \text{ cm} + 80 \text{ cm}) \]

Calculating \(r + h\):

\[ 7 \text{ cm} + 80 \text{ cm} = 87 \text{ cm} \]

Now substituting back into the TSA formula:

\[ \text{TSA} = 2 \times \frac{22}{7} \times 7 \times 87 \]

Again, the \(7\) cancels out:

\[ \text{TSA} = 2 \times 22 \times 87 \]

Calculating:

\[ \text{TSA} = 44 \times 87 = 3828 \text{ cm}^2 \]

### Final Results:
- Volume of the cylinder: \(12320 \text{ cm}^3\)
- Curved Surface Area: \(3520 \text{ cm}^2\)
- Total Surface Area: \(3828 \text{ cm}^2\)

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