Sectors and Arc Length of a Circle Cheat Sheet

Key Facts

• The circumference of a circle is $C = 2\pi r$, where $r$ is the radius.
• The area of a circle is $A = \pi r^2$, where $r$ is the radius.
• Arc length in degrees is $s = \frac{\theta}{360}\cdot 2\pi r$, where $\theta$ is the central angle in degrees.
• Sector area in degrees is $A = \frac{\theta}{360}\cdot \pi r^2$, where $\theta$ is the central angle in degrees.
• Arc length in radians is $s = r\theta$, where $\theta$ is measured in radians.
• Sector area in radians is $A = \frac{1}{2}r^2\theta$, where $\theta$ is measured in radians.
• To convert degrees to radians, use $\theta_{\text{rad}} = \theta_{\text{deg}}\cdot \frac{\pi}{180}$.
• To convert radians to degrees, use $\theta_{\text{deg}} = \theta_{\text{rad}}\cdot \frac{180}{\pi}$.

Vocabulary

Circle

A circle is the set of all points in a plane that are the same distance from a fixed center point.

Radius

The radius is the distance from the center of a circle to any point on the circle.

Arc

An arc is a connected part of a circle's circumference.

Central Angle

A central angle is an angle whose vertex is at the center of the circle and whose sides are radii.

Sector

A sector is the region of a circle bounded by two radii and the arc between them.

Radian

A radian is an angle measure where $1$ radian subtends an arc length equal to the radius.