Areas Related to Circles
In this section, we delve into understanding the sector and segment of a circle. A sector of a circle is defined as the area enclosed by two radii and the arc connecting them, while a segment is defined as the area enclosed by a chord and the arc that connects its endpoints. We categorize sectors as minor and major based on the angle subtended at the circle's center, while a corresponding minor and major segment are defined similarly.
To calculate the area of a sector, we use the formula:
$$
\text{Area of sector} = \frac{θ}{360} \times πr^2
$$
Here, \(θ\) is the angle in degrees, and \(r\) is the radius of the circle. The length of the arc can also be determined with:
$$
\text{Length of arc} = \frac{θ}{360} \times 2πr
$$
Next, to find the area of a segment, we subtract the area of the triangle formed by the two radii from the area of the corresponding sector:
$$
\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}
$$
The section concludes with practical examples that illustrate these principles, fostering an understanding of how to apply these formulas in real-world contexts.