## kamuela710 3 years ago If the measure of an exterior angle of a regular polygon is 24, how many sides does the polygon have?

1. GT

exterior angle? Can you draw which one it is?

2. moneybird

360/n = 24

3. moneybird

n is the number of sides

4. MrYantho

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. GT

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

6. moneybird

i think exterior angle is 180 - interior angle

7. moneybird

|dw:1327116266165:dw|

8. GT

How can that be 24 for a polygon?

9. MrYantho

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

10. moneybird

It's 24 sides

11. GT

See that drawing helped.

12. GT

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. MrYantho

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. GT

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

15. MrYantho

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

16. GT

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. MrYantho

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

18. GT

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