Geometry Sum Of Polygon Interior Angles
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Add Those Angles
Sum Of Polygon Interior Angles

 

 The angles inside any triangle add up to 180 degrees.

 

 
 What about the angles inside a shape with four corners?
 

 

 
 How come the angles inside a triangle add up to 180 degrees
 and the angles inside a shape with 4 corners add up to 360 degrees?
 
 It works like this ...
 
 Think of the line that draws the shape like a sidewalk.
 In any shape, 
 from some starting point you go one time around a circle
 and end up back where you started ...
 

 

 
 One complete turn around equals 360 degrees.
 Yeah, but the triangle has only 180 degrees.
 Just hold on, I'm not done yet ...
 
 To make the explanation easier,
 we'll use a triangle with all the same angles 
 but it will work with any triangle.
 
 So say you're walking along this sidewalk on the triangle 
 and you come to the first angle (turn).
 

 

 
 Since there are three angles in the triangle,
 we turn 1/3 of the whole 360 each time we turn.
 

 

 
 So we turned 120 degrees.
 

 

 
 Now figure out what's left on the inside ...
 

 

 
 So there's 60 degrees left on the inside.
 60 degrees times three turns is 180 degrees.
 And there you are.
 
 OK, now let's try a 4 sided shape.
 To keep things simple, we'll use a square.
 But this will work with any 4 sided shape ...
 

 

 
 Now try a pentagon (5 sided).
 

 

 
 OK, let's stop a second and see what's happening here ...
 
Number of Sides Total of Interior Angles
3 180  
4 360   (180 + 180 = 360)
5 540   (360 + 180) = 540
 
 Every time we add another side,
 the interior angles add up to 180 degrees more.
 
 That means we can continue this chart
 as far as we want really fast.
 
Number of Sides Total of Interior Angles
3 180  
4 360  
5 540  
6 720  
7 900  
8 1080  
 
 The more sides (and angles) we have in an object,
 the more the angles inside the shape add up to!
 
 We could even write a formula like this ...
 

 Angles inside a polygon add up to = 180 (number of sides - 2)

 
 If we just kept adding more and more sides
 until the number of sides gets to infinity,
 what shape would we have?
 

 A CIRCLE!

 

   copyright 2005 Bruce Kirkpatrick

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