Find the area of the minor segment of a circle of radius 28 cm, when the angle of the corresponding sector is `45^@.`

Answer

To find the area of the minor segment of a circle with a radius of 28 cm and a central angle of 45 degrees, we can follow these steps:

### Step 1: Calculate the Area of the Sector
The area of a sector of a circle can be calculated using the formula:

\[ \text{Area of Sector} = \frac{\pi r^2 \theta}{360} \]

Where:
- \( r \) is the radius of the circle
- \( \theta \) is the angle of the sector in degrees

Substituting the values:

\[ \text{Area of Sector} = \frac{\pi \times (28)^2 \times 45}{360} \]

### Step 2: Simplify the Area of the Sector
Calculating \( 28^2 \):

\[ 28^2 = 784 \]

Now substituting this back into the area formula:

\[ \text{Area of Sector} = \frac{\pi \times 784 \times 45}{360} \]

### Step 3: Calculate the Area of the Triangle
The area of triangle OAC can be calculated using the formula:

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

In this case, both the base and height can be derived from the radius and the angle. The height can be calculated using:

\[ \text{Height} = r \sin(\theta) \]

Substituting the values:

\[ \text{Height} = 28 \sin(45^\circ) = 28 \times \frac{1}{\sqrt{2}} = \frac{28}{\sqrt{2}} = 14\sqrt{2} \]

Now, the area of triangle OAC becomes:

\[ \text{Area of Triangle} = \frac{1}{2} \times 28 \times 14\sqrt{2} \]

### Step 4: Calculate the Area of the Minor Segment
The area of the minor segment can be found by subtracting the area of the triangle from the area of the sector:

\[ \text{Area of Minor Segment} = \text{Area of Sector} - \text{Area of Triangle} \]

### Step 5: Substitute and Simplify
Now, substituting the calculated areas into the equation:

1. Area of Sector:

\[ \text{Area of Sector} = \frac{\pi \times 784 \times 45}{360} = \frac{35280\pi}{360} = 98\pi \text{ cm}^2 \]

2. Area of Triangle:

\[ \text{Area of Triangle} = \frac{1}{2} \times 28 \times 14\sqrt{2} = 196\sqrt{2} \text{ cm}^2 \]

Now substituting these into the area of the minor segment:

\[ \text{Area of Minor Segment} = 98\pi - 196\sqrt{2} \]

### Final Calculation
Using approximate values for \(\pi\) and \(\sqrt{2}\):

\[ \text{Area of Minor Segment} \approx 98 \times 3.14 - 196 \times 1.414 \]

Calculating:

\[ \approx 307.72 - 277.44 \approx 30.28 \text{ cm}^2 \]

### Final Answer
Thus, the area of the minor segment is approximately:

\[ \text{Area of Minor Segment} \approx 30.28 \text{ cm}^2 \]

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