Geometry More Arc Lengths
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Same Old, Same Old
More Arc Lengths

 

 

 
 Remember from the page about crossed lines
 that angles A and C are the same size,
 and angles B and D are the same size ...
 
 We can drop a circle over these crossed lines
 so the center of the circle is right at the cross.
 

 

 
 Just dropping the circle on this puppy 
 doesn't change the angles.
 The measure of angle A
 is still the same as the measure of angle C.
 And the measure of angle B
 is still the same as the measure of angle D.
 
 We put the center of the circle where the lines crossed.
 That means, that the piece of the circle out from angle A
 is the same length as the piece of the circle out from angle C!
 

 

 
 Since the piece of the circle out from angle A
 is equal to the piece of the circle out from angle C,
 it should be no big shock that the piece of the circle
 out from angle B is equal to the piece of the circle
 out from angle D!
 

 

 
 This may seem kind of boring,
 but math types do lots of stuff with this
 
 Example:
 

 

 
 The diameter of the circle is 10 
 The measure of angle D is 40 degrees
 What is the length of the circle out from angle A?
 
 Since angle D is 40, angle B is 40 also ...
 

 

 
 Now we need to find the measure of angle A ...
 
 The angles add up to 360 degrees
 Angle A is the same as angle C.
 
 So:

 A + 40 + C + 40 = 360

A + A + 80 = 360

2A + 80 = 360

2A = 280

A = 140

 

 

 
 So we need to know the arc length 
 out from the 140 degree angle on top.
 
 Way back when we said that the diameter was 10.
 That makes the circumference 10 x p
 
 The arc on top is 140 of the 360 so:
 

Arc Length = 10p  

140

360

 Arc Length 12.217

 

   copyright 2005 Bruce Kirkpatrick

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