If the measure of an exterior angle of a regular polygon is 24, how many sides does the polygon have?

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- anonymous

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- anonymous

exterior angle? Can you draw which one it is?

- anonymous

360/n = 24

- anonymous

n is the number of sides

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- anonymous

To do this, we have to remember two things:
1. The exterior angles of a polygon ALWAYS add up to 360 degress
2. A regular polygon's angles are all equal (including exterior)
So, every exterior angle is 24
\[n=360/24\]
\[n=15\]

- anonymous

How is it then called exterior angle? That is the angle between two adjacent vertices and the center of the polygon.

- anonymous

i think exterior angle is 180 - interior angle

- anonymous

|dw:1327116266165:dw|

- anonymous

How can that be 24 for a polygon?

- anonymous

|dw:1327116301791:dw|
Interior and exterior angles sum to 180

- anonymous

It's 24 sides

- anonymous

See that drawing helped.

- anonymous

OK. Then, the interior angle is 156. So, the angle at the center of the polygon must be 24 also. Therefore, 360/24 sides. Got it.

- anonymous

Nope. There are 15 sides (see earlier answer). Exterior angles always sum to 360 no matter how many sides. So it is n=360/24
The sum of interior angles is different depending on the number of sides:
\[\sum=(n-2)*180\] where n is the number of sides.

- anonymous

360/24 = 15. What is the issue there?

- anonymous

Sorry, that was a misread by me. :-P

- anonymous

BTW - I hate that "formula" thing with n-2 thing. It is unnecessary. Basic triangle property of angles and the circle being 360 degrees does the trick here, and then it is all basic arithmetic.

- anonymous

I completely agree. It's just a much longer explanation for a forum like this.

- anonymous

Yeah. But, I like calculating knowing geometric properties rather than having people memorize a formula.

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