Question

WILL AWARD MEDAL/FAN!
How can you find the number of sides of an equiangular polygon by measuring one of its interior angles? By measuring one of its exterior angles?

Thank you for your help! It is greatly appreciated!~

Answer 1

I really don't like dealing with the majority of geometry students here, but you're lucky. This is a straightforward question. Are you familiar with equiangular polygons?

Answer 2

No, not really. I'm not really sure what they are. Can you please explain or draw it for me?

Answer 3

According to traditional euclidean theory, equiangular polygons are figures that are at complete equilibrium. For example, a square would be an equiangular polygon as all its angles and sides are equal.

Answer 4

The interior of any shape is derived from a circle. According to Pythagoras, a circle can technically (for this purpose) be defined as having an angle of "360". If you had a square, all of it's angles would be 90. 90*4=360

Answer 5

Same for a triangle. If you cut a square in half, it's equivalent to cutting a circle in half. If you were able to tear and shape that triangle back into a circle, you would only have an angle of 180.

Answer 6

Do you understand?

Answer 7

oh, so does that mean a rectangle can also be an equiangular polygon? also a rhombus?

Answer 8

@happyyanee4 ?

Answer 9

sorry for the late replies. cpu crashed )):

Answer 10

A rhombus is only equiangular of it is a square.

Answer 11

Do you know the calculation of the angles as determined by the number of sides?

Answer 12

happyyanee4 gave you two clues to find the calculation.

If sides are 4, then 4 * 90 = 360
If sides are 3, then 3 * 60 = 180

So both the angle and the number of sides change at the same time.

Answer 13

so the answer is you have to know the number of sides first?

Answer 14

No.

Answer 15

i'm confused.

Answer 16

Well, you know a 3 sided opbject happens to have a total of 180 degrees and that an equiangular version is 60 per angle. You likewise know that a 4 sided object happns to contain 360 segrees, but each angle is 90. So no matter ehat the total is, the individual angles in an equiangular object will be some number and that number changes.

Answer 17

Wait so how do i describe the answer?

Answer 18

Well, you have to describe that. And you have to know how to do it in relationship to both the internal and external angles.

Answer 19

so you just say like "you have to know the degree of an angle"...?

Answer 20

i don't know how to explain that relationship between the angle measure and the number of sides

Answer 21

Well, there is a formula, which I asked if you knew. This one:
http://www.regentsprep.org/regents/math/geometry/gg3/lpoly2.htm

Answer 22

because if it was the other way around, you could find the measure with the number of sides, instead of finding the number of sides with a measure.

Answer 23

yes, that was the one i was talking about ^
yes, i am familiar with it

Answer 24

wait so basically, the answer to this problem is just the reverse of that formula?

Answer 25

basically you need to revise it to find the number of sides.

Answer 26

the reverse?

Answer 27

Well, there is going to be a similar relationship. How much work you need to put into the answer depends on if they want a formula or an idea. Oh, and here is a neat one... you don't have to do much for the second question.

Answer 28

the second question's answer is:
by knowing the interior angle?

Answer 29

basically, yah. If you answer the first one, you can just describe the relationship of the exterior to interior angles. Then reference the first answer.

Answer 30

yeah.
the only problem?
LOL we don't know the answer to the first one. well, that is, not really.

Answer 31

Well, 3s=60, and 4s=90.... lets see if those solve to the same number.

Answer 32

do you mean proportions?

Answer 33

20 and ... nope... so something is missing from the easy method.

Answer 34

no, they don't.

Answer 35

yeah

Answer 36

Hmm... but that formula used n-2... so that will be involved somehow. And I bet if you looked you could find the formula for finding the n of a n gon from the angle. Just trying to cook one up will take a little more work.

Answer 37

180 (n-2) = n
^ formula for finding n? ^

Answer 38

Yah, probably easiest to just solve that for n.

Answer 39

but n could be anything, since n is an unknown polygon, right?

Answer 40

But you would need the sum of interiaor angles... hmmm.... so might still have another thing to solve.

Answer 41

i'm sorry but i'm so lost right now. like the first question's answer is just use the formula to figure out the number of sides, right?

Answer 42

No. Because you don't know the sum.

Answer 43

oh yeah....oops

Answer 44

Hmmm... geometric sequence.

Answer 45

then how do we solve this problem?

Answer 46

what do we need to do with the formula?

Answer 47

Well, using the formula I found a 5 sided... 540 degrees... but that just tells me it is a geometric sequence. Not sure where happy was going with the circle.

Answer 48

what?

Answer 49

What is I am thinking rather than googling.

Answer 50

wait me?
i'm not googling anything LOL
does it sound like i'm not trying? it's probably cuz i'm hecka tired. it's like three in the morn here. sorry. but i'm trying, so that i don't waste your time and mine. but math is just sooo not my subject.

Answer 51

No, no. I am just thinking out loud rather than googling. So what I type is incomplete.

Answer 52

oh psh.

Answer 53

See, while the angles are getting larger, the difference btween each angle and the previous one is getting smaller, so not a simple sequence. A fraction in there...

Answer 54

Hmmm... what if I did not work with interior? Exterior?

Answer 55

OK, stuff I was playing with on a scrach pad...

s=180(n-2)

s/180=(n-2)

s/180 +2=n

3 180 60 + 30
4 360 90 + 18
5 540 108 + 12
6 720 120 +
7 900 not a nice number...


a_n - a_1 * r^{n-1}
180/n^2
270/n^2

Sides Ex-Ang
3 120
4 90
5 72
6 120


Most of it is junk, but I just noticed something.

The 3 sided and 6 sided.

3 sides has an interior of 60 and exterior of 120.
6 sides has an interior of 120 and exterior of 60.

To me, that suggests there is a numeric relationship.

Answer 56

but other sides, like 5 and 7 don't really have a relationship...

Answer 57

yah, but I know I am looking for a fraction, and it would be 5 and 10 to look at.

Answer 58

So if I try interiors...

x/3 = 60
x/6 = 120

Ewww... that won't work.

Exteriors:
x/3 = 120
x/6 = 60

3*120 = 360
6*60 = 360

Oooh.... And there is that circle happy talked about!

Answer 59

So try a few more with the exterior angle. Divide 360 by the exterior angle and see if you get the number of sides.

Answer 60

oh i see
sort of

Answer 61

8 sides = 1080 degrees, or 135 per angle, which is 45 on the exterior. That matches up nicely with the 90 on a 4 sided. =)

360/90 = 4
360/45 = 8

Answer 62

So the key is what happy said, a circle as 360 degrees, and what the question asked about, interior and \(\textbf{exterior}\) angles.

Answer 63

okay, thank you so much!!~

Answer 64

you helped greatly! it's four now, and i must be getting some sleep. thank you!