Sectors

A sector is a pie-shaped segment of a circle formed by connecting a circle’s center with line segments to two points on the circle itself, as shown in the diagram below.

Here,

• $r$ is the radius of the circle,
• $\theta$ is the angle between the two line segments (see Greek letters)
• $s$ is the arclength of the sector
• $A$ is the area of the sector

Arclength and sector area formulas

The arclength $s$ and sector area $A$ are each related to $r$ and $\theta$ through the formulas in the table below. The choice of formula to use depends on if you are measuring $\theta$ in degrees or radians.

$\theta$ is measured in…	Arclength formula	Sector area formula
degrees	$s=\frac{\pi }{180^{\circ }}\cdot \theta r$	$A=\frac{\pi }{360^{\circ }}\cdot \theta r^2$
radians	$s=\theta r$	$A=\frac{1}{2}\theta r^2$

The formulas in radians are worth memorizing.

Why do we like radians so much in mathematics?

The relative simplicity of the formulas shown above when $\theta$ is measured in radians is one of the primary reasons why radians is so heavily used in mathematics in the precalculus level and beyond. The choice of using radians yields significant simpler formulas in the field of calculus when dealing with trigonometric functions.