Find the volume, curved surface area and the total surface area of a cone having base radius 35 cm and height 84 cm.

Answer

To find the volume, curved surface area, and total surface area of a cone with a base radius of 35 cm and height of 84 cm, we can follow these steps:

### Step 1: Calculate the Slant Height (L)
The formula for the slant height \( L \) of a cone is given by:
\[ L = \sqrt{R^2 + H^2} \]
where \( R \) is the radius and \( H \) is the height.

Given:
- Radius \( R = 35 \) cm
- Height \( H = 84 \) cm

Calculating \( L \):
\[ L = \sqrt{35^2 + 84^2} = \sqrt{1225 + 7056} = \sqrt{8281} = 91 \text{ cm} \]

### Step 2: Calculate the Curved Surface Area (CSA)
The formula for the curved surface area of a cone is:
\[ \text{CSA} = \pi R L \]
Using \( \pi \approx \frac{22}{7} \):
\[ \text{CSA} = \frac{22}{7} \times 35 \times 91 \]

Calculating:
\[ \text{CSA} = \frac{22 \times 35 \times 91}{7} = 22 \times 5 \times 91 = 10010 \text{ cm}^2 \]

### Step 3: Calculate the Total Surface Area (TSA)
The formula for the total surface area of a cone is:
\[ \text{TSA} = \pi R (L + R) \]
Substituting the values:
\[ \text{TSA} = \frac{22}{7} \times 35 \times (91 + 35) \]

Calculating:
\[ \text{TSA} = \frac{22}{7} \times 35 \times 126 \]
\[ \text{TSA} = \frac{22 \times 35 \times 126}{7} = 22 \times 5 \times 126 = 13860 \text{ cm}^2 \]

### Step 4: Calculate the Volume (V)
The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi R^2 H \]
Substituting the values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 35^2 \times 84 \]

Calculating:
\[ V = \frac{1}{3} \times \frac{22}{7} \times 1225 \times 84 \]
\[ V = \frac{22 \times 1225 \times 84}{3 \times 7} = \frac{22 \times 1225 \times 28}{1} = 107800 \text{ cm}^3 \]

### Summary of Results
- Curved Surface Area (CSA) = 10010 cm²
- Total Surface Area (TSA) = 13860 cm²
- Volume (V) = 107800 cm³