Find the area of a segment of a circle of radius 2 km , if the arc of the segment has a measure of `60^(@)` .

Answer

To find the area of the segment of a circle with a radius of 2 km and an arc measuring 60 degrees, we can follow these steps:

### Step 1: Find 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 and \( \theta \) is the angle in degrees.

Given:
- Radius \( r = 2 \) km
- Angle \( \theta = 60^\circ \)

Substituting the values into the formula:
\[ \text{Area of Sector} = \frac{\pi \times (2)^2 \times 60}{360} \]
\[ = \frac{\pi \times 4 \times 60}{360} \]
\[ = \frac{240\pi}{360} \]
\[ = \frac{2\pi}{3} \text{ km}^2 \]

### Step 2: Find the area of triangle OAC
Since the triangle OAC is formed by two radii and the chord, it is an isosceles triangle. Given that the angle at O is \( 60^\circ \), we can find the area of triangle OAC using the formula for the area of an isosceles triangle:
\[ \text{Area} = \frac{1}{2} \times b \times h \]
However, since we know it's an equilateral triangle (as all angles are \( 60^\circ \)), we can use the formula for the area of an equilateral triangle:
\[ \text{Area} = \frac{\sqrt{3}}{4} a^2 \]
where \( a \) is the length of a side.

Here, \( a = 2 \) km, so:
\[ \text{Area of Triangle OAC} = \frac{\sqrt{3}}{4} \times (2)^2 \]
\[ = \frac{\sqrt{3}}{4} \times 4 \]
\[ = \sqrt{3} \text{ km}^2 \]

### Step 3: Find the area of the segment
The area of the segment can be found by subtracting the area of triangle OAC from the area of the sector:
\[ \text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle OAC} \]
Substituting the areas we found:
\[ \text{Area of Segment} = \frac{2\pi}{3} - \sqrt{3} \]

### Step 4: Approximate the area of the segment
Using the approximate values \( \pi \approx 3.14 \) and \( \sqrt{3} \approx 1.732 \):
\[ \text{Area of Segment} \approx \frac{2 \times 3.14}{3} - 1.732 \]
\[ \approx \frac{6.28}{3} - 1.732 \]
\[ \approx 2.093 - 1.732 \]
\[ \approx 0.361 \text{ km}^2 \]

### Final Answer
The area of the segment is approximately \( 0.361 \text{ km}^2 \).

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