Areas of Sector and Segment of a Circle
In this section, we analyze the concepts of sectors and segments within circles, which are vital in understanding circular areas. A sector is defined as the portion of a circle enclosed by two radii and the respective arc. Conversely, a segment lies between a chord and the arc corresponding to that chord.
Key Concepts:
- Minor and Major Sectors: The smaller sector formed is called the minor sector, while the larger one is referred to as the major sector. The angle of the major sector is defined as the total circle (360°) minus the angle of the minor sector.
- Calculating the Area of a Sector: The area can be calculated by the formula:
$$\text{Area of sector} = \frac{\theta}{360} \times \pi r^2$$
where \(\theta\) is the angle at the center in degrees and \(r\) is the radius.
- Arc Length: The formula for the arc length is:
$$\text{Length of arc} = \frac{\theta}{360} \times 2\pi r$$
- Segments of a Circle: The area of a segment is derived by subtracting the area of the associated triangle (formed by the radii and the chord) from the area of the sector.
$$\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}$$
Practical Examples and Exercises:
Through examples and exercises, students reinforce their understanding by calculating areas for given sectors and segments under different scenarios, solidifying their grasp on circular areas.