Definition:Sector

Definition

A sector of a circle is the area bounded by two radii and an arc.

As Euclid defined it:

A sector of a circle is the figure which, when an angle is constructed at the center of the circle, is contained by the straight lines containing the angle and the circumference cut off by them.

(The Elements: Book III: Definition $10$)

In the diagram below, $BAC$ is a sector.

In fact there are two sectors, together making up the whole of the circle.

When describing a sector by means of its three defining points (as in $BAC$ above), the convention is to report them in the following order:

$(1):\quad$ The point on the circumference at the end of the clockwise radius

$(2):\quad$ The point at the center of the circle

$(3):\quad$ The point on the circumference at the end of the anticlockwise radius

Thus in the sector above, $BAC$ describes the sector indicated by $\theta$, while the sector comprising the rest of the circle is described by $CAB$.

Angle of Sector

The angle of the sector is the angle between the two radii which delimit the sector.

In the above diagram, the angle of the sector $BAC$ is $\theta$.

Again, the conjugate angle of $\theta$ also forms a sector, denoted $CAB$ (see above).