# Definition:Sector

## Definition

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

In the words of Euclid:

*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 $\text{III}$: 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 a sector** is the angle between the two radii which delimit the sector.

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

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