Circles and Sectors

Arcs, Chords, Sectors, and Segments

A circle is a set of all points in a plane that are equidistant from a fixed point, called the center, denoted as point . This is represented as circle .
  • The point is the center of the circle.
  • A line segment connecting the center to any point on the circle is called the radius.
When two points and are chosen on circle , the circle is divided into two parts, each called an arc. The arc connecting and is denoted as .
  • By default, refers to the shorter arc. To represent the longer arc, an additional point on the arc is included, denoted as .
A straight line passing through two points on the circle is called a secant, and the line segment between these two points is called a chord. The chord connecting and is denoted as chord .
  • A chord passing through the center of the circle is the diameter.
A figure formed by an arc and two radii, and , is called a sector, denoted as sector .
  • In sector , is called the central angle corresponding to arc .
A figure formed by a chord and the arc it subtends, , is called a segment of the circle.

Properties of Sectors

In a given circle, two sectors with the same central angle are congruent. Therefore:
  • The arcs corresponding to equal central angles have the same length.
  • The areas of the sectors corresponding to equal central angles are equal.
In a single circle, if the central angle of a sector is doubled, tripled, and so on, the arc length and the area of the sector also double, triple, and so on.
Thus, the arc length and area of a sector are directly proportional to the measure of its central angle.

Key Properties of Sectors

In a single circle or congruent circles:
  1. Sectors with equal central angles have equal arc lengths and areas.
  2. The arc length and area of a sector are directly proportional to its central angle.