In this section, we delve into important formulas related to circles found in geometry. The length of an arc in a sector can be calculated using the formula:
- Length of an Arc:
\\[\\text{Length of Arc} = \\frac{\\theta}{360} \\times 2\\pi r \\]
where \\( r \\) is the radius of the circle and \\( \\theta \\) is the angle measure in degrees.
We also examine the area of a sector of a circle defined by its radius and the angle at the center. The formula for the area of a sector is:
- Area of a Sector:
\\[\\text{Area of Sector} = \\frac{\\theta}{360} \\times \\pi r^2 \\]
Finally, the area of a segment is derived by subtracting the area of the triangle formed by the radii and the chord from the area of the sector:
- Area of Segment:
\[\text{Area of Segment} = \text{Area of Sector} - \text{Area of Triangle} \]
These principles are crucial for applications in fields such as architecture, engineering, and any spatial analyses involving circular shapes.