The height of a cone is 16 cm and its base radius is 12 cm, then what is the curved surface area and the total surface area of cone ? $(\pi = 3.14)$

Question

Accepted Answer

To determine the curved surface area and the total surface area of the cone, we need to use the formulas for these measurements.

1. **Curved Surface Area (CSA) of a Cone:**
   The formula for the curved surface area of a cone is given by:
   $$
   \text{CSA} = \pi r l
   $$
   where $ r $ is the radius of the base and $ l $ is the slant height of the cone. The slant height $ l $ can be calculated using the Pythagorean theorem:
   $$
   l = \sqrt{r^2 + h^2}
   $$
   where $ h $ is the height of the cone.

Given:
   - Height $ h = 16 $ cm
   - Radius $ r = 12 $ cm

Calculate the slant height $ l $:
   $$
   l = \sqrt{12^2 + 16^2} = \sqrt{144 + 256} = \sqrt{400} = 20 \text{ cm}
   $$

Now, calculate the curved surface area:
   $$
   \text{CSA} = \pi \times 12 \times 20 = 3.14 \times 12 \times 20 = 753.6 \text{ cm}^2
   $$

2. **Total Surface Area (TSA) of a Cone:**
   The formula for the total surface area of a cone is given by:
   $$
   \text{TSA} = \pi r (r + l)
   $$

Calculate the total surface area:
   $$
   \text{TSA} = \pi \times 12 \times (12 + 20) = 3.14 \times 12 \times 32 = 1205.76 \text{ cm}^2
   $$

Based on these calculations, the curved surface area is $ 753.6 \text{ cm}^2 $ and the total surface area is $ 1205.76 \text{ cm}^2 $. Therefore, Option 1 is correct.

**Explanation for Incorrect Options:**

- **Option 2:** Lists the total surface area first and the curved surface area second, which is incorrect based on the order specified in the question.
- **Option 3:** Incorrectly states the total surface area as $ 1210.76 \text{ cm}^2 $, which does not match the calculated value of $ 1205.76 \text{ cm}^2 $.
- **Option 4:** Incorrectly states the curved surface area as $ 452.16 \text{ cm}^2 $, which does not match the calculated value of $ 753.6 \text{ cm}^2 $.

Thus, Option 1 is the only correct choice, as it accurately reflects the calculated values for both the curved surface area and the total surface area of the cone.