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An arc of a circle is a "portion" of the circumference of the circle.
The length of an arc is simply the length of its "portion" of the circumference. The circumference itself can be considered a full circle arc length. |
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Arc Measure: In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc. |
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Arc Length: In a circle, the length of an arc is a portion of the circumference.
The letter "s" is used to represent arc length. |
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Consider the following proportion:
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If we solve the proportion for arc length, and replace "arc measure"
with its equivalent "central angle", we can establish the formula:
Notice that arc length is a fractional part of the circumference. For example, an arc measure of 60º is one-sixth of the circle (360º), so the length of that arc will be one-sixth of the circumference of the circle.
In circle O, the radius is 8 inches and minor arc  is intercepted by a central angle of 110 degrees. Find the length of minor arc to the nearest integer.
As you progress in your study of mathematics and angles, you will see more references made to the term "radians" instead of "degrees". So, what is a "radian" ?
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The radian measure, θ, of a central angle is defined as the ratio of the length of the arc |
the angle subtends, s, divided by the radius of the circle, r. |
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which gives arc length, s: s = θr
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subtend = "to be opposite to"
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One radian is the central angle that subtends an
arc length of one radius (s = r).
Since all circles are similar, one radian is the same value for all circles. |
Relationship between Degrees and Radians:
In a circle, the arc measure of the entire circle is 360º and the arc length of the entire circle is represented by the formula for circumference of the circle:  .
Substituting C into the formula s = θr shows:
C = θr 2πr = θr 2π = θ
The arc measure of the central angle of an entire circle is 360º and the radian measure of the central angle of an entire circle = 2π.
360º (degrees) = 2π (radians)
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To change
from degrees to radians,
multiply degrees by  |
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To change
from radians to degrees,
multiply radians by  |
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Justify why: The length of the arc intercepted by a central angle is proportional to the radius. |
The diagram at the right shows two circles with the same center (concentric circles). It has already been shown that concentric circles are similar under a dilation transformation.
The ratio of similitude of the smaller circle to the larger circle is:
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| The same dilation that mapped the smaller circle onto the larger circle will also map the slice (sector) of the smaller circle with an arc length of s1 onto the slice (sector) of the larger circle with an arc length of s2. When the radius gets dilated by a scale factor, the arc length is also dilated by that same scale factor. |
 As long as the central angles are the same, the slices (sectors) will be similar. |
Since corresponding parts of similar figures are in proportion, 
An equivalent proportion can be written as  This proportion shows that the ratio of the arc length intercepted by a central angle to the radius of the circle will always yield the same (constant) ratio.
In relation to the two arc length formulas seen on this page, both show that arc length, s, is expressed as "some value" times the radius, r. The arc length is proportional to the radius.
When θ is in degrees:
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When θ is in radians:
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Setting r = 1 shows the constant of proportionality.
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to degrees.
to degrees.
radians in a circle of radius 20 centimeters.
