Circle Properties & Arc Measures Cheat Sheet

Key Facts

• The circumference of a circle is $C = 2\pi r$ or $C = \pi d$, where $r$ is the radius and $d$ is the diameter.
• The area of a circle is $A = \pi r^2$.
• The diameter is twice the radius, so $d = 2r$ and $r = \frac{d}{2}$.
• A full circle has angle measure $360^\circ$, so a semicircle has measure $180^\circ$.
• The measure of a minor arc equals the measure of its central angle, so if $\angle AOB = 70^\circ$, then $m\widehat{AB} = 70^\circ$.
• Arc length is $s = \frac{\theta}{360^\circ}\cdot 2\pi r$ when the central angle $\theta$ is measured in degrees.
• Sector area is $A_{\text{sector}} = \frac{\theta}{360^\circ}\cdot \pi r^2$ when the central angle $\theta$ is measured in degrees.
• An inscribed angle measures half its intercepted arc, so $m\angle ABC = \frac{1}{2}m\widehat{AC}$.

Vocabulary

Circle

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

Radius

A radius is a segment from the center of a circle to any point on the circle.

Diameter

A diameter is a chord that passes through the center of the circle and has length $d = 2r$.

Chord

A chord is a segment whose endpoints both lie on the circle.

Arc

An arc is a connected part of a circle measured in degrees or by its length along the circle.

Sector

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