The total surface area of a cone of radius 3 cm and height 4 cm is ________.

Answer

To find the total surface area of a cone with a radius of 3 cm and a height of 4 cm, we will follow these steps:

### Step 1: Identify the formula for the total surface area of a cone.
The total surface area (TSA) of a cone is given by the formula:
\[ \text{TSA} = \pi r (r + l) \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 2: Calculate the slant height using the Pythagorean theorem.
The slant height \( l \) can be calculated using the formula:
\[ l = \sqrt{h^2 + r^2} \]
where \( h \) is the height of the cone.

Given:
- Radius \( r = 3 \) cm
- Height \( h = 4 \) cm

Substituting the values:
\[ l = \sqrt{4^2 + 3^2} = \sqrt{16 + 9} = \sqrt{25} = 5 \text{ cm} \]

### Step 3: Substitute the values into the TSA formula.
Now we have:
- \( r = 3 \) cm
- \( l = 5 \) cm

Substituting these values into the TSA formula:
\[ \text{TSA} = \pi \times 3 \times (3 + 5) = \pi \times 3 \times 8 \]

### Step 4: Calculate the total surface area.
Now calculate:
\[ \text{TSA} = \pi \times 3 \times 8 = 24\pi \text{ cm}^2 \]

### Final Answer:
The total surface area of the cone is \( 24\pi \) cm². If you want a numerical approximation, using \( \pi \approx \frac{22}{7} \):
\[ \text{TSA} \approx 24 \times \frac{22}{7} \approx 75.43 \text{ cm}^2 \]