The radius and vertical height of a cone are 5 cm and 12 cm, respectively. Then, its lateral surface area

Options

A. `204.3 cm^2`
B. `205.4cm^2`
C. `200.3cm^2`
D. None of these

Answer

To find the lateral surface area of a cone with a radius of 5 cm and a vertical height of 12 cm, we will follow these steps:

### Step 1: Identify the given values
- Radius (r) = 5 cm
- Vertical height (h) = 12 cm

### Step 2: Find the slant height (l)
The slant height of the cone can be found using the Pythagorean theorem. In a right triangle formed by the radius, height, and slant height, we have:

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

Substituting the known values:

\[ l = \sqrt{(5)^2 + (12)^2} \]
\[ l = \sqrt{25 + 144} \]
\[ l = \sqrt{169} \]
\[ l = 13 \text{ cm} \]

### Step 3: Use the formula for lateral surface area
The formula for the lateral surface area (LSA) of a cone is given by:

\[ \text{LSA} = \pi r l \]

### Step 4: Substitute the values into the formula
Using \( \pi \approx \frac{22}{7} \):

\[ \text{LSA} = \frac{22}{7} \times 5 \times 13 \]

### Step 5: Calculate the lateral surface area
Now performing the multiplication:

\[ \text{LSA} = \frac{22 \times 5 \times 13}{7} \]
\[ \text{LSA} = \frac{1430}{7} \]
\[ \text{LSA} = 204.2857 \text{ cm}^2 \]

### Step 6: Round the answer
Rounding to one decimal place, we get:

\[ \text{LSA} \approx 204.3 \text{ cm}^2 \]

### Final Answer
The lateral surface area of the cone is approximately **204.3 cm²**.

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