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.
arc
definition
Arc Measure: In a circle, the degree measure of an arc is equal to the measure of the central angle that intercepts the arc.

definition
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.            
arcpics
Consider the following proportion:

proportionarc

If we solve the proportion for arc length, and replace "arc measure"
with its equivalent "central angle", we can establish the formula:

arc length2

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.

example
In circle O, the radius is 8 inches and minor arc arcac is intercepted by a central angle of 110 degrees. Find the length of minor arc arcac to the nearest integer.
    
exampleANS2

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Radian Measure

 


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" ?

definition
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.

radianform   
 
which gives arc length, s:     s = θr

arctheta
subtend
subtend = "to be opposite to"
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.

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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: circumformulas.

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)

unitcircle
360º = 2π
(divide both sides by 2)
180º = π
360º = 2π
(divide both sides by 4)
9-
360º = 2π
(divide both sides by 360)
180
360º = 2π
(divide both sides by 2π)
1rad
visual
To change
from degrees to radians,
multiply degrees by changedr
To change
from radians to degrees,
multiply radians by changerd


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exs

1. Convert 60º to radians.
    radex1
2. Convert 135º to radians.
    radex2ans

3. Convert radex3to degrees.
    radex3ans
4. Convert radex4to degrees.
   radex4ans

5. Find the length of an arc subtended by an angle of 7pi4radians in a circle of radius 20 centimeters.
    ex5ans

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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:
smallerlarger

arclengthcircle
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.
slicesAs long as the central angles are the same, the slices (sectors) will be similar.

Since corresponding parts of similar figures are in proportion, circleproportion
An equivalent proportion can be written as circleproportion2 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:

arclength1

When θ is in radians:

arclength22

Setting r = 1 shows the constant of proportionality.

radius3

radiou5


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