if each interior angle of a regular polygon measures 144 degrees, how many sides does the polygon have?

- anonymous

- katieb

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- anonymous

Please show steps on how to solve.

- anonymous

obtuse angles but idont know about a polygon with obtuse angles

- 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?

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## More answers

- anonymous

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

- anonymous

decagon?

- JamesJ

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

- anonymous

ok

- JamesJ

What's the sum of angles of a square?

- anonymous

180 (4-2) =
360

- anonymous

360 ofcourse.as all angles are 90

- anonymous

the polygon in question has 10 sides

- anonymous

^ why?

- 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|

- anonymous

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

- 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
\[ \frac{180(n-2)}{n} \]

- JamesJ

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

- anonymous

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

- anonymous

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

- anonymous

NOPE THATS NOT RIGHT ^

- anonymous

i dont understnad..

- 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?

- anonymous

OH WOW
I GET IT!

- anonymous

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

- JamesJ

...hence
\[ 180n - 360 = 144n \]
Now?

- anonymous

expand the brackets hellobudy

- anonymous

nope.. now what?

- JamesJ

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

- anonymous

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

- JamesJ

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

- anonymous

k i think i get it..

- anonymous

thank you!

- 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.

- anonymous

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

- anonymous

ok ill try with 100 degrees

- JamesJ

No, that won't work. One sec.

- JamesJ

try 108 degrees

- anonymous

ALRIGHT :)

- anonymous

5 sides?

- JamesJ

yes

- anonymous

:D

- JamesJ

last one, 162 degrees

- anonymous

20 :)

- JamesJ

what about for 180 degrees?

- anonymous

huh?

- JamesJ

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

- anonymous

idk waht your asking

- JamesJ

It would look like a circle.

- 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

- anonymous

thanks :)

- anonymous

g2g

- JamesJ

Here's an even better picture of this idea:

##### 1 Attachment

- anonymous

thanks!

- anonymous

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

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