The radius of a right circular cone is 3 cm and its height is 4 cm. The total surface area of the cone is

Options

A. 48.4 sq. cm
B. 64.4 sq. cm
C. 96.4 sq. cm
D. 75.4 sq. cm

Answer

To find the total surface area of a right circular cone, we can use the formula:

\[ \text{Total Surface Area} = \pi r l + \pi r^2 \]

where:
- \( r \) is the radius of the base,
- \( l \) is the slant height of the cone.

### Step 1: Identify the given values
The radius \( r \) is given as 3 cm, and the height \( h \) is given as 4 cm.

### Step 2: Calculate the slant height \( l \)
To find the slant height \( l \), we can use the Pythagorean theorem:

\[ l = \sqrt{r^2 + h^2} \]

Substituting the known values:

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

### Step 3: Substitute values into the total surface area formula
Now we can substitute \( r \) and \( l \) into the total surface area formula:

\[ \text{Total Surface Area} = \pi r l + \pi r^2 \]

Substituting \( r = 3 \) cm and \( l = 5 \) cm:

\[ \text{Total Surface Area} = \pi (3)(5) + \pi (3^2) \]

### Step 4: Simplify the expression
This simplifies to:

\[ \text{Total Surface Area} = 15\pi + 9\pi = 24\pi \]

### Step 5: Calculate the numerical value
Using \( \pi \approx \frac{22}{7} \):

\[ \text{Total Surface Area} = 24 \times \frac{22}{7} = \frac{528}{7} \text{ cm}^2 \]

Calculating this gives:

\[ \text{Total Surface Area} \approx 75.43 \text{ cm}^2 \]

### Final Answer
The total surface area of the cone is approximately \( 75.43 \text{ cm}^2 \).

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