Question

1) Show that the exterior angle of a regular dodecagon (12 sides) is 30 degrees
2) The size of an interior angle in a regular polygon is 175 degrees. How many sides has the polygon?

I've tried and I have no idea how to answer these questions!

Answer 1

exterior angle = 360/12 = 30

Answer 2

I know that but if it says 'show', don't I need a proof? I have no idea how to prove it

Answer 3

The sum of the interior angles of an n-sided polygon is (n-2) times 180. Can you assume that?

Answer 4

The question doesn't say. That is exactly what it says in the textbook

Answer 5

its been a long time - but i don't remember seeing a proof - though there must be one i suppose

Answer 6

Well, start by assuming that it's true. Then you have 1800 degrees total divided by 12 = 150. That's the measure of each interior angle in a regular dodecagon. The exterior angles are the supplementary angles to that 180 - 150 = 30.

Answer 7

The proof is a classic, by induction. I'll find a link.

Answer 8

So do you think it would be okay to just do 360/12?

Answer 9

Yes. Another way to look at it is to imagine yourself walking around the perimeter of this polygon. At each turn you will turn the same angle (say to the left if going ccw). The turns are all the same size. When you finish you will have turned 360 degrees. 360/12 = 30.

Answer 10

I am still a bit confused as to how I am supposed to 'show' this. There isn't any proof for this, right?

Answer 11

its obviously true for a rectangle - so does that mean its true for every polygon - just a thought

Answer 12

The sum of the interior angles of an n-sided polygon is (n-2) times 180. Can you assume that? If not, we can talk about how to prove it by induction.

Answer 13

I am not sure what you mean by 'assume'

Answer 14

"Assume" means that the formula 180 x (n-2) is known to be true.

Answer 15

Ah, yes. We can assume that

Answer 16

OK. I am going to go out on a limb here and say that you can start with the formula. Then by the calculation above you get 150 for the interior angles and 180 - 150 for the exterior angles.

Answer 17

The second problem has interior angles of 175. So the exterior angle is 5. How many times do you have to turn by 5 degrees to do a total of 360?

Answer 18

35?

Answer 19

360/5 = ?

Answer 20

Ah, 72

Answer 21

Sounds right to me

Answer 22

one way to think about first part might be that when the polygon 'grows' in number of sides to infinite number of sides - this closely approaches a circle - which is 360 degrees.

Answer 23

Oh, okay

Answer 24

i don't know if I've explained that very well!

Answer 25

You did :)

Answer 26

Dont' know if this is simple enough to help you but here is a link about the induction proof
http://answers.yahoo.com/question/index?qid=20061227163133AAH25J7

Answer 27

Ok. Thanks everyone!

Answer 28

yw

Answer 29

yea - thats a good proof