The largest triangle is inscribed in a semi-circle of radius 4 cm. Find the area inside the semi-circle which is not occupied by the triangle.

Options

A. `8(pi - 2) sq cm`
B. `7(pi - 1)sq cm`
C. `6(pi- 2) sq cm`
D. None of these

Answer

To solve the problem of finding the area inside the semicircle that is not occupied by the largest triangle inscribed in it, follow these steps:

### Step 1: Calculate the area of the semicircle.
The formula for the area of a semicircle is given by:

\[ \text{Area of semicircle} = \frac{1}{2} \pi r^2 \]

Given that the radius \( r \) is 4 cm, we can substitute this value into the formula:

\[ \text{Area of semicircle} = \frac{1}{2} \pi (4)^2 = \frac{1}{2} \pi (16) = 8\pi \text{ cm}^2 \]

### Step 2: Determine the area of the largest triangle inscribed in the semicircle.
The largest triangle that can be inscribed in a semicircle is an isosceles triangle with its base on the diameter of the semicircle. The height of this triangle is equal to the radius of the semicircle.

The base of the triangle is equal to the diameter of the semicircle, which is:

\[ \text{Base} = 2r = 2 \times 4 = 8 \text{ cm} \]

The height of the triangle is equal to the radius:

\[ \text{Height} = r = 4 \text{ cm} \]

Now, we can calculate the area of the triangle using the formula:

\[ \text{Area of triangle} = \frac{1}{2} \times \text{base} \times \text{height} \]

Substituting the values we found:

\[ \text{Area of triangle} = \frac{1}{2} \times 8 \times 4 = \frac{1}{2} \times 32 = 16 \text{ cm}^2 \]

### Step 3: Calculate the area inside the semicircle not occupied by the triangle.
To find the area inside the semicircle that is not occupied by the triangle, we subtract the area of the triangle from the area of the semicircle:

\[ \text{Area not occupied} = \text{Area of semicircle} - \text{Area of triangle} \]

Substituting the areas we calculated:

\[ \text{Area not occupied} = 8\pi - 16 \text{ cm}^2 \]

### Final Answer:
The area inside the semicircle which is not occupied by the triangle is:

\[ 8\pi - 16 \text{ cm}^2 \]