A central angle is an angle with its vertex at the center of a circle:

a.  If the central angle cuts off a quarter-circle, what is the measure of the central angle?
b.  If the central angle cuts off a semicircle, what is the measure of the central angle?
c.  If the central angle cuts off one-third of a circle, what is the measure of the central angle?
d.  Find a general rule for central angles based on how much of the circle they cut off.

 

 

 
Answers:  
a.  The measure is 90 (or 360 / 4).
b.  The measure is 180 (or 360 / 2).
c.  The measure is 120 (or 360 / 3).
d.  If a central angle cuts off an arc of one n-th of the full circle, its measure is
360 / n.

 

  Central Angle:  
A central angle is an angle formed by two intersecting radii such that its vertex is at the center of the circle.
 

Central Angle = Intercepted Arc

<AOB is a central angle.
 Its intercepted arc is the minor arc from A to B.  m<AOB = 80
 

  Inscribed Angle:
An inscribed angle is an angle with its vertex "on" the circle, formed by two intersecting chords.

Inscribed Angle = Intercepted Arc

<ABC is an inscribed angle.
 Its intercepted arc is the minor arc from A to C.
m<ABC = 50