Find the volume , curved surface area and the total surface area of a hemisphere of radius 7 cm .

Answer

To find the volume, curved surface area, and total surface area of a hemisphere with a radius of 7 cm, we can follow these steps:

### Step 1: Find the Volume of the Hemisphere
The formula for the volume \( V \) of a hemisphere is given by:
\[ V = \frac{2}{3} \pi r^3 \]
Where \( r \) is the radius.

**Calculation:**
- Substitute \( r = 7 \) cm into the formula:
\[ V = \frac{2}{3} \times \pi \times (7)^3 \]
- Calculate \( 7^3 = 343 \):
\[ V = \frac{2}{3} \times \pi \times 343 \]
- Use \( \pi \approx \frac{22}{7} \):
\[ V = \frac{2}{3} \times \frac{22}{7} \times 343 \]
- Simplify:
\[ V = \frac{2 \times 22 \times 343}{3 \times 7} \]
- Calculate \( 2 \times 22 = 44 \):
\[ V = \frac{44 \times 343}{21} \]
- Now calculate \( 44 \times 343 = 15192 \):
\[ V = \frac{15192}{21} \approx 723.42857 \text{ cm}^3 \]
- Thus, the volume of the hemisphere is approximately \( 723.43 \text{ cm}^3 \).

### Step 2: Find the Curved Surface Area of the Hemisphere
The formula for the curved surface area \( CSA \) of a hemisphere is given by:
\[ CSA = 2 \pi r^2 \]

**Calculation:**
- Substitute \( r = 7 \) cm into the formula:
\[ CSA = 2 \times \pi \times (7)^2 \]
- Calculate \( 7^2 = 49 \):
\[ CSA = 2 \times \pi \times 49 \]
- Use \( \pi \approx \frac{22}{7} \):
\[ CSA = 2 \times \frac{22}{7} \times 49 \]
- Simplify:
\[ CSA = \frac{2 \times 22 \times 49}{7} \]
- Calculate \( 2 \times 22 = 44 \):
\[ CSA = \frac{44 \times 49}{7} \]
- Now calculate \( 44 \times 49 = 2156 \):
\[ CSA = \frac{2156}{7} \approx 308 \text{ cm}^2 \]
- Thus, the curved surface area of the hemisphere is \( 308 \text{ cm}^2 \).

### Step 3: Find the Total Surface Area of the Hemisphere
The formula for the total surface area \( TSA \) of a hemisphere is given by:
\[ TSA = 3 \pi r^2 \]

**Calculation:**
- Substitute \( r = 7 \) cm into the formula:
\[ TSA = 3 \times \pi \times (7)^2 \]
- We already calculated \( 7^2 = 49 \):
\[ TSA = 3 \times \pi \times 49 \]
- Use \( \pi \approx \frac{22}{7} \):
\[ TSA = 3 \times \frac{22}{7} \times 49 \]
- Simplify:
\[ TSA = \frac{3 \times 22 \times 49}{7} \]
- Calculate \( 3 \times 22 = 66 \):
\[ TSA = \frac{66 \times 49}{7} \]
- Now calculate \( 66 \times 49 = 3234 \):
\[ TSA = \frac{3234}{7} \approx 462 \text{ cm}^2 \]
- Thus, the total surface area of the hemisphere is \( 462 \text{ cm}^2 \).

### Final Results:
- Volume of the hemisphere: \( 723.43 \text{ cm}^3 \)
- Curved surface area: \( 308 \text{ cm}^2 \)
- Total surface area: \( 462 \text{ cm}^2 \)