A sector in a circle is defined as the region enclosed by two radii and the arc that connects them.

To visualize this concept, think of a pizza. When you cut a slice of pizza, you are effectively creating a sector. Mathematically, a sector is formed by two radii (the plural of radius) extending from the center of the circle to its circumference, along with the arc that lies between these two radii. The radii are straight lines, while the arc represents the curved segment of the circle’s boundary that joins them.

The size of a sector can vary significantly. When the two radii are positioned closely together, the resulting sector resembles a thin slice of pizza. Conversely, if the radii are spaced farther apart, the sector will be larger, akin to a substantial slice of pizza. The angle formed between the two radii, measured in degrees, determines the sector’s size and is known as the central angle.

For instance, if the central angle measures $90^\circ$, the sector represents one-quarter of the circle, since $90$ degrees is one-quarter of $360$ degrees (the total degrees in a complete circle). Similarly, a central angle of $180^\circ$ results in a sector that is a semicircle, or half of the circle.

Understanding sectors is crucial in various mathematical fields, including geometry and trigonometry. They play a significant role in calculating areas and arc lengths, which have practical applications in fields such as engineering, architecture, and even in everyday activities like slicing a cake!