Definition: The angle that an arc makes at the center of the circle of which it is a part.
Try this Drag one of the orange dots that define the endpoints of the blue arc.
Note how the arc angle changes.
One measure of an arc is the angle formed by the arc at the center of the circle that it is a part of.
(The other is the length of the arc  see Length of an Arc.)
In the figure above, click 'reset' and note that the angle measure of the arc BA is 60°.
To see how it derived, click 'Show central angle', and note that the 60° is the angle made by the arc at the center of the circle.
This angle measure is written like this:
and is read as "the measure of arc AB is 60 degrees".
On a diagram
When arc angle measures are marked on a diagram, there are two common ways to do it:
1. Write the angle alongside the arc itself.
This is less cluttered, but be sure to add the degree mark or it may get confused with the arc length.
In the diagram above click 'reset' to see this form.


2. You can draw the lines from the arc endpoints to the center point and label the central angle in the usual way.
In the diagram above, click 'Show central angle' to see this form.


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Other circle topics
General
Equations of a circle
Angles in a circle
Arcs
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