## hellobuddy 3 years ago if each interior angle of a regular polygon measures 144 degrees, how many sides does the polygon have?

1. hellobuddy

Please show steps on how to solve.

2. jesusisrisen

obtuse angles but idont know about a polygon with obtuse angles

3. JamesJ

Ok. What's the sum of angles of a triangle? 180 degrees. A triangle is a polygon with 3 sides. What's the sum of the angles of a square? A square is a polygon with 4 sides. What's the answer here?

4. hellobuddy

the answer is supposed to be 10 sides how do you solve that?

5. jesusisrisen

decagon?

6. JamesJ

We'll get there ... follow me in logic first

7. hellobuddy

ok

8. JamesJ

What's the sum of angles of a square?

9. hellobuddy

180 (4-2) = 360

10. Safiah

360 ofcourse.as all angles are 90

11. Safiah

the polygon in question has 10 sides

12. hellobuddy

^ why?

13. JamesJ

Right ... so you've got the formula for an n-sided polygon. It has a sum of angles $180(n-2)$ and that's because you can make a polygon out of triangles. Stick two triangles together and you have a square, hence the sum of their angles is 180 + 180. Take a square add a triangle and you have a pentagon.|dw:1327439866826:dw|

14. hellobuddy

i know how to do that, how do you solve when you dont know the number of sides?

15. JamesJ

Now, if an n-sided polygon has a sum of angles 180(n-2), what's the size of the angles if each of them is the same? Well, there are n angles, hence one angle has the size of $\frac{180(n-2)}{n}$

16. JamesJ

For your problem, set that equal to 144 and solve for n.

17. Safiah

the general rule is Each Angle (of a Regular Polygon) = (n-2) × 180° / n so 144=(n-2)x180/n 144n=(n-2)x180 144n=180n-360 144n-180n=-360 36n=360 n=10

18. hellobuddy

so, James J, i do 180 (n-2) / n = 144 144-180+2= n/n?

19. hellobuddy

NOPE THATS NOT RIGHT ^

20. hellobuddy

21. JamesJ

Safiah has written this out for you. But to start it off again $\frac{180(n-2)}{n} = 144$ hence $180(n-2) = 144n$ Can you do it now?

22. hellobuddy

OH WOW I GET IT!

23. hellobuddy

sooo 180( n-2) = 144n wiat no, what do i do now?

24. JamesJ

...hence $180n - 360 = 144n$ Now?

25. Safiah

expand the brackets hellobudy

26. hellobuddy

nope.. now what?

27. JamesJ

Subtract 144n from both sides... $180n - 144n - 360 = 0$ i.e. $36n - 360 = 0$ Now can you finish it?

28. hellobuddy

-360 = -36 and then you divide 360 by 36 and you get ten

29. JamesJ

right, $36n = 360$ hence $n = \frac{360}{36} = 10$

30. hellobuddy

k i think i get it..

31. hellobuddy

thank you!

32. JamesJ

Do yourself a favor. Take a blank piece of paper. Write out the solution again. When you can do that without looking at anything --such as this web or another version of the solution--then you know you understand the solution.

33. hellobuddy

ok! can you give me another example so i can practice? use a different degrees?

34. hellobuddy

ok ill try with 100 degrees

35. JamesJ

No, that won't work. One sec.

36. JamesJ

try 108 degrees

37. hellobuddy

ALRIGHT :)

38. hellobuddy

5 sides?

39. JamesJ

yes

40. hellobuddy

:D

41. JamesJ

last one, 162 degrees

42. hellobuddy

20 :)

43. JamesJ

44. hellobuddy

huh?

45. JamesJ

That would correspond to a regular polygon with an infinite number of sides. What does that look like?

46. hellobuddy

47. JamesJ

It would look like a circle.

48. JamesJ

As you add more and more sides to a regular polygon, it looks more and more like a circle. In the limit, as the number of sides goes to infinity, it would become a circle. http://www.ck12.org/ck12/images?id=301055

49. hellobuddy

thanks :)

50. hellobuddy

g2g

51. JamesJ

Here's an even better picture of this idea:

52. hellobuddy

thanks!

53. cshalvey

Great job JamesJ - and way to hang in there hellobuddy!