A regular n-gon is inscribed in the unit circle.
What is the perimeter for each n below?
a.3
b.5
c.6
d.10
e.57
f.542
g. n
h. The perimeter in part f should be close to what
number? How close is it?

- metalslayer

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- schrodinger

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- anonymous

Aye, mate. I may be of some assistance. But first, I need a favor from ye. Tell me when the observatory is.

- metalslayer

What observatory?

- metalslayer

I really need help on this

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## More answers

- geerky42

Do you know Trigonometry? Or at least Law of Cosines?

- geerky42

If so, this site may helps: http://mathcentral.uregina.ca/QQ/database/QQ.09.07/h/lindsay2.html

- metalslayer

I've been on that website. I just don't get it. Can you please just solve for A and show the steps so I could do the rest? Thanks.

- metalslayer

I do know trig and law of cosines and sines

- geerky42

What A?

- metalslayer

a.3
b.5
c.6
d.10
e.57
f.542
g. n
h. The perimeter in part f should be close to what
number? How close is it?

- metalslayer

Please show me how to answer some of them with the steps. Thank You.

- metalslayer

By A i meant n=3

- geerky42

From site, our equation would be \(P = nc = n\sqrt{2-2\cos\left(\dfrac{360^\text o}{n}\right)}\)

- geerky42

Do you understand how this equation was derived?

- metalslayer

P is perimeter, n is the n-gon, what is c?

- geerky42

c is side.

- metalslayer

How do I find the side? I am only given the n.

- geerky42

You can use law of cosines to find the length of sides:|dw:1436579931950:dw|

- geerky42

n-gon is inscribed in unit circle, which has radius of 1.

- metalslayer

So lets try to solve a.) n=3. I will plug in n into 360/n so it will equal 120 degrees. Then I have no idea lol.

- geerky42

Yes, Plugging in n=3, you would have \(P = 3\sqrt{2-2\cos\left(120^\text o\right)}\)

- geerky42

You can just use calculator.

- metalslayer

p=5.2?

- geerky42

Yeah I got approximately that.

- metalslayer

So first I divide 360 by n, and then plug it into the formula?

- geerky42

Actually, you just plug in whether value of n, then calculate.

- geerky42

For case of n=3, you just plug in n=3 and calculate \(P = 3\sqrt{2-2\cos\left(\dfrac{360^\text o}{3}\right)}\)

- metalslayer

Oh I see. Thank you. What about if n = 542?

- geerky42

Same. Plug in n=542 then calculate \(P = \textbf{542}\sqrt{2-2\cos\left(\dfrac{360^\text o}{\textbf{542}}\right)}\)

- metalslayer

Great, thanks. And lastly could you please explain part h.

- metalslayer

h. The perimeter in part f should be close to what
number? How close is it?

- geerky42

Well, imagine if n goes to infinity, what would n-gon become into?

- metalslayer

a circle

- geerky42

Right. Saying we have case of \(n=\infty\). How can we find perimeter?

- geerky42

Or should I say "circumference"?

- metalslayer

\[\pi radius squared\]

- metalslayer

pi radius squared

- geerky42

That's area of circle. We want circumference.

- metalslayer

2pi*r

- metalslayer

so what is the radius?

- geerky42

Yeah. Here, we have UNIT circle, so radius is 1.
So part f should be close to \(2\pi\)

- metalslayer

So that's my answer? Thank You. You are so helpful.

- geerky42

Yeah. Though I am not sure how to answer "How close is it?"
I guess subtract part f from \(2\pi\).

- geerky42

But I think you did your job good enough lol...

- metalslayer

Thank you. This helped me so much.

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