Sector

Regions Bounded by Two Radii and an Arc

A sector is the region of a circle bounded by two radii and one arc. A slice cut from the center of a round cake is a useful model: its two straight sides represent radii, and its curved edge represents the arc.

Every sector has three main components that are interconnected. First are the two radii that meet at the center of the circle. Second is the arc that connects the endpoints of both radii. Third is the area enclosed by both radii and the arc.

Characteristics of Sectors

A sector has its vertex at the circle's center. The two radii meet there and form the central angle, while the connecting arc completes the boundary.

For a fixed circle, the central angle determines what fraction of the circle belongs to the sector. A larger central angle selects a larger fraction. A smaller angle selects a smaller fraction.

If the chosen central angle is less than $180^\circ$, the bounded region is the minor sector. If it is greater than $180^\circ$, the region is the major sector. An angle of exactly $180^\circ$ gives a semicircle.

Within the same circle, arc length and sector area are both proportional to the central angle. Consequently, equal central angles in that circle determine equal arc lengths and equal sector areas.

Relationship Between Central Angle and Sector

The central angle is measured at the circle's center between the two bounding radii. It can be expressed in degrees or radians.

For example, $90^\circ$ is one quarter of a full turn, so its sector has one quarter of the circle's area. A $180^\circ$ sector has half of the circle's area.

The relationship is a part-to-whole ratio. If the central angle is $\theta$ in degrees, then the ratio of sector area to circle area is $\frac{\theta}{360^\circ}$. If $\theta$ is measured in radians, the ratio is $\frac{\theta}{2\pi}$.

Recognizing Sectors in Context

Sector geometry can describe a region swept through a fixed angle. For example, an idealized sprinkler with range $r$ and sweep angle $\theta$ covers a sector before obstacles, wind, and uneven water distribution are considered.

Sectors also appear in circular diagrams and radial templates. The central-angle ratio determines the corresponding fraction of the full disk, but geometry alone does not determine a machine's load or efficiency.

On an evenly marked clock face, adjacent hour marks bound one of $12$ equal sectors, each with a central angle of $30^\circ$. A slice cut from the center of a round pizza or cake is another familiar model, provided its sides follow radii.