The largest triangle is inscribed in a semi-circle of radius 7 cm. Find the area inside the semi-circle which is not occupied by triangle.

Answer

To find the area inside the semi-circle that is not occupied by the largest triangle inscribed in it, we can follow these steps:

### Step 1: Calculate the area of the semi-circle.
The formula for the area of a semi-circle is given by:
\[ \text{Area of semi-circle} = \frac{1}{2} \pi r^2 \]
Given the radius \( r = 7 \) cm, we can substitute this value into the formula:
\[ \text{Area of semi-circle} = \frac{1}{2} \pi (7)^2 = \frac{1}{2} \pi \times 49 = \frac{49\pi}{2} \text{ cm}^2 \]
Using \( \pi \approx 3.14 \):
\[ \text{Area of semi-circle} \approx \frac{49 \times 3.14}{2} \approx \frac{153.86}{2} \approx 76.93 \text{ cm}^2 \]

### Step 2: Calculate the area of the triangle.
The largest triangle that can be inscribed in a semi-circle is a right triangle, where the base is equal to the diameter of the semi-circle and the height is equal to the radius.

- The diameter \( d = 2r = 2 \times 7 = 14 \) cm (this is the base of the triangle).
- The height \( h = r = 7 \) cm.

The area of the triangle can be calculated using the formula:
\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values we have:
\[ \text{Area of triangle} = \frac{1}{2} \times 14 \times 7 = \frac{1}{2} \times 98 = 49 \text{ cm}^2 \]

### Step 3: Calculate the area inside the semi-circle that is not occupied by the triangle.
To find the area not occupied by the triangle, we subtract the area of the triangle from the area of the semi-circle:
\[ \text{Area not occupied} = \text{Area of semi-circle} - \text{Area of triangle} \]
Substituting the areas we calculated:
\[ \text{Area not occupied} = 76.93 \text{ cm}^2 - 49 \text{ cm}^2 = 27.93 \text{ cm}^2 \]

### Final Answer:
The area inside the semi-circle which is not occupied by the triangle is approximately:
\[ \text{Area not occupied} \approx 27.93 \text{ cm}^2 \]