## anonymous 4 years ago If the measure of an exterior angle of a regular polygon is 24, how many sides does the polygon have?

1. anonymous

exterior angle? Can you draw which one it is?

2. anonymous

360/n = 24

3. anonymous

n is the number of sides

4. 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 $n=360/24$ $n=15$

5. anonymous

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

6. anonymous

i think exterior angle is 180 - interior angle

7. anonymous

|dw:1327116266165:dw|

8. anonymous

How can that be 24 for a polygon?

9. anonymous

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

10. anonymous

It's 24 sides

11. anonymous

See that drawing helped.

12. 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.

13. 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.

14. anonymous

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

15. anonymous

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

16. 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.

17. anonymous

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

18. anonymous

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