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Our first journey into angles associated with circles will be the central angle.
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A central angle of a circle is an angle formed by two radii with the vertex at the center of the circle. |
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In circle O, at the right, ∠AOB is a central angle.
The portion of the circle in red is called a "minor arc AB".
The remainder of the circle is called "major arc AB".
An arc is a portion of the circle's circumference.
 Major arcs are usually named using three letters to distinguish them from minor arcs.
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• Minor Arc: -
any arc whose length is less than the length of a semicircle (with a corresponding central angle less than 180º).
• Major Arc: -
any arc whose length is greater than the length of a semicircle (with a corresponding central angle greater than 180º).
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The degree measure of a minor arc of a circle is the measure of the corresponding central angle.  |
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• The degree measure of a semicircle is 180.
• The degree measure of a major arc is equal to 360 minus the degree measure of its corresponding minor arc.
| Measuring arcs:
• The degree of an arc refers to the angle degree measure of the arc's corresponding central angle.
• The length of an arc refers to the distance along the curve of the arc, determined by a linear measure (number of feet, inches, etc) taken up by the arc's portion of the circumference.
In the lessons on "angles", the "degree of an arc" will be used.
The "length of an arc" will be investigated in the lesson Arc Length. |
  but
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The definition of the degree measure of a minor arc. stated above,
gives us our formula for measuring a central angle in a circle.
Central Angle (definition of degree measure of arc)
The measure of a central angle equals
the measure of its intercepted arc.
Central Angle = Intercepted Arc
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In the diagram at the right, ∠AOB is a central angle with an intercepted minor arc from A to B.
The m∠AOB = the measure of the intercepted arc.
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m∠ AOB = 82º |
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An "intercepted arc" is the arc "cut off" by the central angle's rays.
The "arc" is between where the rays of the angle cross the circle.
(See more about terminology at the end of this page.)
Example 1:
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Solution:
• ∠AOB is a central angle, therefore the measure of its intercepted arc is also 60º.
• The diameter creates an arc of 180º from point A through B to point C.
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Example 2:
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Solution:
• The central angle ∠AOB = 76º.
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Due to the radii sides, ΔAOB is an isosceles triangle, where the base angles at points A and B will be congruent.
x + x + 76 = 180º
2x + 76 = 180
2x = 104
x = 52º
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Theorems Associated with Central Angles:
In a circle, or congruent circles, congruent central angles have congruent arcs.
(the converse is also true) |
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In a circle, or congruent circles, congruent central angles have congruent chords.
(the converse is also true) |
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Terminology: "intercepted" and "subtended":
When working with arcs and circles in geometry, two descriptors are used:
"intercepted" and "subtended".
• An arc is "intercepted" by an angle when the angle's rays pass through the circumference of a circle cutting off an arc. This can also be applied to a chord "intercepting an arc".
• An angle is "subtended" by an arc when lines drawn through the endpoints of the arc intersect to form an angle. This can also be applied to a chord being "subtended by an arc".
In the statementa above, an "angle intercepts an arc" where as an "arc subtends an angle".
While this seems to be the accepted difference in usaage of these two terms,
you will actually see the terms "intercepted" and "subtended" used interchangeably.
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