Find the volume, curved surface area and the total surface area of a cone whose height is 6 cm and slant height 10 cm. (Take `pi = 3.14`)

Explanation

To find the volume, curved surface area, and total surface area of a cone with a height of 6 cm and a slant height of 10 cm, we can follow these steps:

### Step 1: Find the radius of the cone
We know the height (h) and the slant height (l) of the cone. We can use the Pythagorean theorem to find the radius (r) of the cone.

Using the formula:
\[ l^2 = r^2 + h^2 \]

Substituting the values:
\[ 10^2 = r^2 + 6^2 \]
\[ 100 = r^2 + 36 \]
\[ r^2 = 100 - 36 \]
\[ r^2 = 64 \]
\[ r = \sqrt{64} = 8 \text{ cm} \]

### Step 2: Calculate the Volume of the Cone
The formula for the volume (V) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]

Substituting the values:
\[ V = \frac{1}{3} \times 3.14 \times (8^2) \times 6 \]
\[ V = \frac{1}{3} \times 3.14 \times 64 \times 6 \]
\[ V = \frac{1}{3} \times 3.14 \times 384 \]
\[ V = \frac{1207.68}{3} \]
\[ V \approx 402.56 \text{ cm}^3 \]

### Step 3: Calculate the Curved Surface Area of the Cone
The formula for the curved surface area (CSA) of a cone is:
\[ \text{CSA} = \pi r l \]

Substituting the values:
\[ \text{CSA} = 3.14 \times 8 \times 10 \]
\[ \text{CSA} = 3.14 \times 80 \]
\[ \text{CSA} = 251.2 \text{ cm}^2 \]

### Step 4: Calculate the Total Surface Area of the Cone
The formula for the total surface area (TSA) of a cone is:
\[ \text{TSA} = \text{CSA} + \pi r^2 \]

Substituting the values:
\[ \text{TSA} = 251.2 + 3.14 \times (8^2) \]
\[ \text{TSA} = 251.2 + 3.14 \times 64 \]
\[ \text{TSA} = 251.2 + 200.96 \]
\[ \text{TSA} = 452.16 \text{ cm}^2 \]

### Final Answers
- Volume of the cone: \( \approx 402.56 \text{ cm}^3 \)
- Curved Surface Area of the cone: \( 251.2 \text{ cm}^2 \)
- Total Surface Area of the cone: \( 452.16 \text{ cm}^2 \)