## hellobuddy Group Title if each interior angle of a regular polygon measures 144 degrees, how many sides does the polygon have? 2 years ago 2 years ago

1. hellobuddy Group Title

Please show steps on how to solve.

2. jesusisrisen Group Title

obtuse angles but idont know about a polygon with obtuse angles

3. JamesJ Group Title

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 Group Title

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

5. jesusisrisen Group Title

decagon?

6. JamesJ Group Title

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

7. hellobuddy Group Title

ok

8. JamesJ Group Title

What's the sum of angles of a square?

9. hellobuddy Group Title

180 (4-2) = 360

10. Safiah Group Title

360 ofcourse.as all angles are 90

11. Safiah Group Title

the polygon in question has 10 sides

12. hellobuddy Group Title

^ why?

13. JamesJ Group Title

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 Group Title

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

15. JamesJ Group Title

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 Group Title

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

17. Safiah Group Title

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 Group Title

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

19. hellobuddy Group Title

NOPE THATS NOT RIGHT ^

20. hellobuddy Group Title

21. JamesJ Group Title

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 Group Title

OH WOW I GET IT!

23. hellobuddy Group Title

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

24. JamesJ Group Title

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

25. Safiah Group Title

expand the brackets hellobudy

26. hellobuddy Group Title

nope.. now what?

27. JamesJ Group Title

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

28. hellobuddy Group Title

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

29. JamesJ Group Title

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

30. hellobuddy Group Title

k i think i get it..

31. hellobuddy Group Title

thank you!

32. JamesJ Group Title

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 Group Title

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

34. hellobuddy Group Title

ok ill try with 100 degrees

35. JamesJ Group Title

No, that won't work. One sec.

36. JamesJ Group Title

try 108 degrees

37. hellobuddy Group Title

ALRIGHT :)

38. hellobuddy Group Title

5 sides?

39. JamesJ Group Title

yes

40. hellobuddy Group Title

:D

41. JamesJ Group Title

last one, 162 degrees

42. hellobuddy Group Title

20 :)

43. JamesJ Group Title

44. hellobuddy Group Title

huh?

45. JamesJ Group Title

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

46. hellobuddy Group Title

47. JamesJ Group Title

It would look like a circle.

48. JamesJ Group Title

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 Group Title

thanks :)

50. hellobuddy Group Title

g2g

51. JamesJ Group Title

Here's an even better picture of this idea:

52. hellobuddy Group Title

thanks!

53. cshalvey Group Title

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