Handshakes/Fencing and Angles in a regular polygon
I have to remember this all the time for fencing, so I'm going to do this in a fencing context. ]
Notes on terminology:
One round of fencing against a single opponent is called a bout. A pool is a group of usually seven fencers who fence each other in the first round of a tournament to determine where they are ranked for the second part of the tournament
To find out the number of bouts there are in a single pool, apply the equation:
n(n-1)/2, where n is the number of fencers in a pool.
Here's why:
Let's use the seven-fencer pool example, we'll call them fencers 1-7. Each fencer must fence everyone else once, not including themselves, so each fencer fences six bouts. If there are seven fencers, that means one would multiply the number of bout per fencer by the number of fencers, 6*7, or n*(n-1). But, if one were to do this, they would count each bout twice, since one would count fencer 1 vs. fencer 4 and fencer 4 vs. fencer 1, whereas in reality there is only the 1 vs. 4 bout. So, divide n*(n-1) by 2, giving you:
n*(n-1)/2
As for the second part of Casey Blue's question
This will be so hard to do without being able to draw you a picture, but follow me as best as you can. I'm going to use a pentagon in my example supplemented by variables, where n is the number of sides.
Begin with a regular pentagon inscribed in a circle. Then draw radii from each point on the circle to the center. Since it is a regular pentagon, the angles between each radius are equal, so divide 360 by 5 (360/n) to find the central angles made by the radius, which comes out to 72 degrees. Now, what you have is 5 isosceles triangles (since the legs are radii) with non-base angles (what's the proper word?) of 72 degrees. Isolate one triangle for now. To find the sum of the base angles, which will be equal to the interior angles of the pentagon (since one base angle is 1/2 of the measure of the interior angle of the pentagon), subtract 72 from 180 (180-360/n). That gives you an interior angle of 108 degrees. To find the sum of the interior angles, multiply by 5 to get 540 (180n-60).
So, the general formulas are:
For an interior angle
180n-360
For the sum of interior angles
180-360/n
(OMG! I actually did a blog entry!)
Notes on terminology:
One round of fencing against a single opponent is called a bout. A pool is a group of usually seven fencers who fence each other in the first round of a tournament to determine where they are ranked for the second part of the tournament
To find out the number of bouts there are in a single pool, apply the equation:
n(n-1)/2, where n is the number of fencers in a pool.
Here's why:
Let's use the seven-fencer pool example, we'll call them fencers 1-7. Each fencer must fence everyone else once, not including themselves, so each fencer fences six bouts. If there are seven fencers, that means one would multiply the number of bout per fencer by the number of fencers, 6*7, or n*(n-1). But, if one were to do this, they would count each bout twice, since one would count fencer 1 vs. fencer 4 and fencer 4 vs. fencer 1, whereas in reality there is only the 1 vs. 4 bout. So, divide n*(n-1) by 2, giving you:
n*(n-1)/2
As for the second part of Casey Blue's question
This will be so hard to do without being able to draw you a picture, but follow me as best as you can. I'm going to use a pentagon in my example supplemented by variables, where n is the number of sides.
Begin with a regular pentagon inscribed in a circle. Then draw radii from each point on the circle to the center. Since it is a regular pentagon, the angles between each radius are equal, so divide 360 by 5 (360/n) to find the central angles made by the radius, which comes out to 72 degrees. Now, what you have is 5 isosceles triangles (since the legs are radii) with non-base angles (what's the proper word?) of 72 degrees. Isolate one triangle for now. To find the sum of the base angles, which will be equal to the interior angles of the pentagon (since one base angle is 1/2 of the measure of the interior angle of the pentagon), subtract 72 from 180 (180-360/n). That gives you an interior angle of 108 degrees. To find the sum of the interior angles, multiply by 5 to get 540 (180n-60).
So, the general formulas are:
For an interior angle
180n-360
For the sum of interior angles
180-360/n
(OMG! I actually did a blog entry!)
posted by Kevin S. at 3:07 PM
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